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VERITAS
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A
DISSERTATION
ON
Richard Pooler,
Holmesdale.
THE SUMMATION OF INFINITE CONVERGING
SERIES WITH ALGEBRAIC DIVISORS.
EXHIBITING A METHOD NOT ONLY INTIRELY NEW,
BUT MUCH MORE GENERAL THAN ANY OTHER WHICH
HAS HITHERTO APPEARED ON THE SUBJECT.
TRANSLATED FROM THE LATIN OF
An M:
LORGNA,
PROFESSOR OF MATHEMATICS IN THE MILITARY COLLEGE OF VERONA.
WITH ILLUSTRATIVE NOTES AND OBSERVATIONS.
AN
TO WHICH IS ADDED,
APPEND
I X;
Containing all the moſt elegant and ufeful FORMULE which have been
inveſtigated for the Summing of the different Orders of Series; with
various Examples to each.
H.
BY
CLARK E.
neque te ut miretur turba labores,
Contentus paucis lectoribus.
HORAT. SERM. Lib. II. Sat. X.
LONDON:
PRINTED FOR THE AUTHOR; AND SOLD BY MR. MURRAY,
N°
32, IN FLEET-STREET.
M DCC LXXIX.
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CHARLES HUTTON, ESQ.
r
FELLOW OF THE ROYAL SOCIETY,
AND
PROFESSOR OF MATHEMATICS
IN THE
ROYAL MILITARY ACADEMY OF WOOLWICH,
W
This Tranſlation, &c. is inſcribed;
ITH the higheſt Regard to thofe confeffedly great
Abilities in that extenfive Branch of Erudition
wherein he is profeffionally engaged that Candour and
Impartiality which have hitherto diftinguiſhed his Pen in
his various Literary Departments-- and that Readineſs
to encourage even the faintest Endeavours which tend to
the Promoting of Scientific Knowledge Difpofitions
that will raiſe him a far more noble Monument to Pofterity,
than that confpicuous Station in the Mathematic World he
now ſo deſervedly poffeffes; and which even thoſe pitiful,
groveling Sons of Envy who have ſo long been carping
muſt allow, to be only juſt Reward to Superior Merit.
B 2
Theſe
DEDICATION.
Theſe are the Motives which have induced me to prefix
his Name to theſe Sheets, and to take the Liberty of
publicly ſubſcribing myſelf
His much obliged,
and obedient,
i
Humble Servant,
1
t
HENRY CLARKE.
14
[ √ ]
PREFACE.
TH
HE Doctrine of Infinite Series is not only one of the moſt
confiderable, but alſo one of the moſt ancient branches of the
Mathematics; though concealed under various forms and names.
It has its origin in the very nature of things; fince we cannot pof-
fibly aſcertain the properties and relations of magnitudes of whatfo-
ever kind, without calling in its aid in fome view or other in the
compariſon of their dimenfions. Accordingly we firft difcover it
in Euclid's tenth book, when he begins to treat of Incommenfura-
bles; and upon which principle he has alfo given the demonftra-
tions of feveral propofitions in the twelfth book concerning the
circle, and the relative proportions of the fphere, cone and cylin-
der. The fame principle we find purfued alfo by Archimedes in
his Kunλ8 Metenois and other treatifes, under the name of Exhauf-
tions; by which he has demonſtrated ſeveral uſeful propofitions re-
lating to the circle, ellipfe and parabola; as alfo to the cone,
ſphere, ſpheroid, parabolic and hyperbolic conoids, the fpiral
fpaces, &c. having in his tenth and eleventh propofitions of fpiral
ſpaces inveſtigated the fum of all the terms of a progreffion of
fquares, where the roots conſtitute an arithmetical progreffion, and
the common difference is equal to the first term.
In this ftate the Doctrine of Infinite Series remained till the year
1635, when Cavallerius publiſhed his Geometria Indivifibilium, in
which he has conſidered it in a fomewhat different light under the
name of Indivifibles. But the foundation upon which this method
is
vi
PREFACE.
is built feems not to be ftrictly geometrical, as the notion of indivi-
fible parts is hard to be digefted; and in fome cafes it is rather un-
certain, which can only be proved by reducing the demonftration.
to the apogogical form of the ancients. After Cavallerius, and his
pupil Torricellius who alfo published a treatise on this fubject in
1644, the Doctrine of Series was much enlarged and promoted by
feveral famous men in the mathematical way; among theſe were
Des Cartes, Fermat, Huddenius, Huygens, Barrow, and Wallis;
the laſt of whom confiderably improved it, under the title of The
Arithmetic of Infinites, in his treatiſe publiſhed in the year 1656.
In this work, by purfuing a method fimilar to that of Archimedes,
though extended to a much greater length, he fhews how to find
not only the fum of the fquares, but of any powers, or roots what-
foever of an arithmetical progreffion; and thereby compares infi-
nite numbers of curvilinear areas with right-lined figures, and with
each other. Mr. James Gregory in his treatiſe intituled Vera
Circuli, et Hyperbola Quadratura, printed at Padua in 1667, and
the year following re-printed at Venice, and publiſhed with another
piece on the transformation of curves, fhews a peculiar method of
Converging Series, in which he ſuppoſes the terms of the feries to
proceed by pairs, the differences of which continually leffen; and
when the number of terms become infinite, the difference quite va-
niſhes, and the two terms becomes equal; either of which is there-
fore the fum or quantity fought. He gives afterwards another
form of Converging Series confifting of fimple terms, to which the
other may be reduced. And about this time Nicholas Mercator
alſo ſhewed a method of fumming an infinite feries, by which he
accompliſhed the Quadrature of the Hyperbola. But the methods.
of Converging Series given us by the great Sir Ifaac Newton, have
much the advantage of either Gregory's or Mercator's; the inveſti-
gations being much more general, as well as the relations of the
terms of which the feries are compoſed; as may be feen in his
treatiſe
PREFACE.
vii
treatiſe intituled The Method of Fluxions and Infinite Series, tranflated
and publiſhed by Mr. Colfon in 1736; alſo in his Analyfis by Equa-
tions of an Infinite Number of Terms, and his Quadratures, both
tranſlated and publiſhed by Mr. Stewart in 1745. For the method
of Prime and Ultimate Ratios, or Fluxions, far exceeded any thing
that had been done before, both in regard to its extenſive applica-
tion, and the conciſeneſs of its demonftrations: Though it muſt be
acknowledged that, as this method is obviously founded on the
fame principles as that of Exhauftions, being at firft chiefly intro-
duced to avoid the tedious method of demonſtration therein, called
ducens ad abfurdum, the fame objections may therefore be made
againſt both; which I have remarked in my Rationale of Circulating
Numbers. And indeed we may truly conclude with Mr. Stewart,
that in the elaborate works of that penetrating genius Archimedes,
may be diſcovered the foundation not only of the method of Ex-
hauſtions, but alſo of Indivifibles, Infiniteffimals, Increments, Differen-
tials, and even of Prime and Ultimate Ratios, or Fluxions; all which
are evidently nothing more than the Doctrine of Infinite Series
confidered in different views.
After our fecond Archimedes had publiſhed his method of Flux-
ions, and Infinite Series were frequently produced by their applica-
tion to the ſolutions of problems, the fummations of thofe various
kinds of Series began to be particularly attended to; but none were.
found ſo untractable as thoſe propofed to be treated in the following
pages-when the terms are affected with algebraic divifors. For
there is not perhaps in the whole extent of abſtracted mathematics:
a fubject more abftrufe, or upon which the greateſt mathematicians
have more employed their endeavours, than the fummations of
theſe ſpecies of feries. Yet, notwithſtanding thoſe. great lumina-
ries in the mathematical fciences, Leibnits, the Bernoulles, Taylor,
Montmorte, Moivre, Goldbach, Stirling, Mac-Laurin, Nicolas,
Euler, Riccati, Simpſon, and ſeveral others, have particularly dif-
4
cuffed
Vill
PREFACE.
cuffed them, they have ſtill come much fhort of that perfection
they are capable of being brought to. For though it muſt be ac-
knowledged, that our countryman the late Mr. Simpfon has carried
the ſubject much farther in his Mathematical Differtations, than
any other writer before him; yet, whoever compares the method
purſued in this treatife with his, cannot long heſitate in giving our
Author's the preference, both with refpect to the perfpicuity of his
inveſtigations, and the generality of his conclufions.
The principal reafons, Mr. Lorgna obferves, that thefe univer-
fally allowed great men have not fo well fucceeded in this branch
of the analytic art, have been the impropriety and narrowness of
their methods of inveftigating the Formule for the fummations.
For, in order that thefe fpecies of feries may not be incompatible
with a General Expreffion, or Algebraic Sum, it has been thought
by all of them effentially requifite- "That the terms be divided
by two or more fimple factors, conftituting in the feries the fame
arithmetical progreffion *.". But in the following pages it is
plainly fhewn how an infinite feries may be propoſed which hath not
theſe properties, and yet admits of an Algebraic Expreffion for the
fum.——“ That the greatest exponent of the indeterminate quan-
tity in the numerator, be at leaſt two degrees lower than the greateſt
exponent of the fame quantity in the denominator +."This is
alfo clearly proved not to be general, as fome feries having that pro-
perty are ſhewn to have no reſtricted fum; while others which dif-
agree in that reſpect are capable of being fummed, either algebra-
ically, or by the help of the most common quadratures. And a
method is pointed out how to diftinguiſh with certainty, whether
the given ſeries of this order be fummible or not.
* Riccati in Præfat. ad Com. de Ser.
+ Riccati cap. III. pag. 75. Stirlingius Tract. de Sum. Ser. pag. 24. Stone's
Math. Dict. Series. Simpſon's Math. Differt. p. 76.
It
PREFACE.
ix
It has likewiſe been afferted by moft writers on this fubject that,
a ſeries of which the figns are general, or not expreffed, cannot
admit of an algebraic fum; yet, that the fummation may be always
effected when the ſigns are reſtricted, by the aſſiſtance of algebraic
curves; and in ſome particular cafes by tranfcendants. But through
the whole of this treatife the formulæ are general in reſpect to the
figns, the fame expreffion giving the fum of the feries whether
they be conftant or change alternately; obferving to make the figns
in the formulæ correfpond with thofe in the given feries. And
though the figns in the given feries be reftricted, it is plainly proved,
that thoſe ſeries are not generally fummible by any of the methods
of inveſtigation that have yet been given us; but often fail in the
moſt ſimple cafes. And as to the fummations of the Reciprocal
Series of uneven powers in natural numbers, Mr. Lorgna obſerves,
that they have hitherto been ranked among the mathematician's
Defiderata. For though thofe famous men John Bernoulli and
Leonard Euler, had difcovered the fummation of feries of this order
in even powers, by the quadrature of the circle, yet they failed in
the other; having ingenuously confeffed their inability to accom-
plish them*. Theſe are likewife clearly inveftigated in this trea-
tife, and that in a general manner, both for the reciprocals of even
and uneven powers, by the areas of tranfcendant curves. So that,
* Quomodo vero Series tra&tandæ fint fi denominatores terminorum fint numeri naturales
ad dimenfiones impares elevati ex. gr. fi hæc fimpliciſſima proponatur Series 1 +
I
I
+ + + &c. cujus utique fumma eft finita nondum confat per hanc
43
noftram methodum. Invitantur Analista ut defectui fuccurant.
Bernoullius in Difquifit. de Sum. Ser. Vol. IV. pag. 42.
Tantopere jam pertractatæ & inveſtigata funt Series Reciproca Poteftatum numerorum
naturalium, ut vix probabile videatur de iis novi quicquam inveniri poffe. Quicunque
enim de fummis Serierum meditati funt, ii fere omnes quoque in fummas hujufmodi Serie-
rum inquifiverunt, neqne tamen ulla methodo eas idoneo modo exprimere potuerunt.
Eulerus in Differt. de Ser. Tom. VII. Acad. Petropol. edita.
C
I think
7
3
PREFACE.
I think we may fafely conclude with our Author, that nothing
more can now be expected or wifhed for in the Doctrine of Series;
being brought to its greateſt perfection.
It may not be amifs to inform the reader, that I have taken the
liberty through the whole tranflation, to change the Differential
Method (which is almoſt always ufed by foreigners) into the
Fluxional Confideration. But the other fymbols made uſe of which
are not common with us, I have yet retained; as being equally
perfpicuous, more concife, and ftill more convenient to the com-
pofitor. They are fuch as need no explanation; but will immedi-
ately occur to the reader on the first perufal. I have alfo made
fuch corrections and alterations in the whole, as feemed to be ab-
folutely neceffary; and which, I prefume, would have naturally
fuggeſted themſelves to the Author himſelf on a fecond impreffion.
Thoſe inveſtigations and remarks which could not well be intro-
duced into the body of the work, I have given by way of notes at
the end of each ſection. Theſe are intended for the younger clafs
of readers; but may not, perhaps, be found unworthy the atten-
tion of the more experienced.
As an appendix to this treatife, I thought nothing could be more
proper, than a collection of the principal Theorems and Formulæ
for the Summation of Series, which are found fcattered in different
authors. For this, I imagine, there needs no apology, as the uti-
lity of fuch a collection, properly exemplified, muſt be obvious to
every one, who is the leaſt converfant in mathematical or philofo-
phical difquifitions. There is one thing however, which may be
thought by fome to ftand in need of an apology, and that is the
introducing of a new term into the fubject; but as this is evidently
done for the fake of brevity, the fame expreffion occurring fo
often, it is hoped it will be excuſed. Thoſe who do not like the
word Summatrix may fubftitute for it The General Sum of the Series
for an indefinite Number of Terms.
:
A DIS-
[xi]
THE
CONTENT S.
SECTION I.
Page
1. Prop. 1.To find the Sum of an Infinite Series, of which the General Term is
I where z denotes the Number of Terms.
p+qz
>
2. Prop. 2. To determine the Summatrix of the preceding Series.
3. Prop. 3. To find the Sum of the fame Series when the Signs are alternately + and —.
To determine the Summatrix of the preceding Series.
4. Prop. 4.
5. Prop. 5.
To find the Sum of the preceding Series when the Denominators are refpec-
tively multiplied by the Terms of the Progreffion m, m³, m³, &c. m
being any affirmative Number not less than 1.
Ib. Prop. 6. To find the Summatrix of the preceding Series.
7. Notes on Section 1.
SECTION II.
15. Prop. 7. To find the Sum of a Series, the Terms of which have a common Numerator,
and the Denominators any Number of Factors in Arith. Progreffion.
16. The Nature of the Expreſſion &c. S iƒ û ƒ å ƒ yš, explained.
17. The General Formula in Prop. 7. applied to the Summing of Series of the Reciprocals
of two fimple Factors in Arith. Progreffion.
19. An Illustration of Prop. 7. by Examples of Series of which the Sums may be expreffed
purely algebraically.
23. Other
xii
13
CONTENTS.
Page
23. Other Examples where the Sums are exhibited by Circular Arcs, or Logarithms.
27. Notes on Section 2.
28. The Manner of investigating the Fluents of the Expreſſions
S
р
I
HSS
IF x
P
Р
I
9
x
X
S
q
I X X X
x
I - X
31. A further Illustration of Prop. 7. by explaining the Manner of Application to the
Formula.
35. Remark on Dr. Halley's Attempt to square the Circle.
38. Form 8th of Emerfon's Fluxions illuftrated.
SECTION III.
41. Prop. 8. To investigate the Summation of Series of the Reciprocals of three fimple
Factors in Arith. Progreffion, by Means of the General Formula in
Prop. 7.
44. Illuftrated by Examples, of which the Summations are expreffed algebraically; among
which are fome that have been hitherto thought incapable of Summation.
47. Other Examples, of which the Sums are obtained by Circular Arcs and Logarithms.
49. Notes on Section 3.
Ib. The Inveſtigation of the Fluent of the Expreffion
S
P
m
I
I
1
$
n
I
n 9
I
x
x
* X
X
n
9
IF X
57. A concife Method of finding the Fluents of certain Expreffions, exemplified in that of
6
نو 220
Ι
I-y²
58. Another Illuftration of Emerſon's 8th Form; also of Simpſon's 5th Prob. p. 371.
Flux, and of Landen's Theorem, p. 99. Lucub.
SECTION IV.
62. Prop. 9. To find the Sums of Series which are the Reciprocals of four Factors in
Arith. Prog. being a further Application of Prop. 7.
65. Examples of pure algebraic Series.
67. Examples of mixt Series.
69. Notes on Section 4.
*64. To
CONTENTS.
xiii
Page
*64. To reduce the Expreſſion, or find the Fluent of
12
S
2
・y². y²° ÿ
1+ y¹²
+ 24
S
¡ — y³ • y'
17
1+ y¹²
12
12
для ў
108 y. y¹9 ÿ.
12
1+ y¹²
SECTION V.
*66. Prop. 10. To find the Sum of a Series, the Numerators of the Terms being in Arith.
Prog. and the Denominators any Number of fimple Factors also in
Arith. Progreffion.
*67. The Summation of Series of which the General Term is
a+bz
p+qz.m+xx.ßz
*69. Examples of algebraic Series.
71. Examples of mixt Series.
72. The Summation of Series of this Form
a+bz
p+qz.m+nz
74. Examples to this Species of Series, pure and mixt.
75. Notes on Section 5.
76. The Investigation of the Fluent of the Expreffion
I
72
S.
b
b
+
9
9
S
P
In
I
9
x
a
D
9
X
x.
b
9.
79. A Geometrical Demonſtration of the Equation
a
a
a
Circ. arc. r
√ 3, t.u +
arc. t.
= arc. t.
arc.t.
2
2
2
3 au
4a+24
SECTION VI.
81. Prop. 11. To find the Sums of Series, of which the General Term is
a+bz
p+qz. m+nz.r+sz
Formula in Prop. 10.
83. Examples to the preceding Form.
87. Notes on Section 6.
; being a further Application of the General
88. Some Miſtakes of an eminent Writer rectified.
91. De Moivre's Method of fumming Series by Multiplication illuſtrated. -
93. Bernoulli's Method of Summation of Series by Subtraction illuſtrated.
{
SECTION
xiv
CONTENT S.
Page
SECTION VII.
95. Prop. 12. To find the Sum of a Series of which the General Form is
a+bz •ctez
p + qz.m+nz.r+sz.t+uz. &c.
96. The Summation of Series, the Numerators of the Terms confifling of two, and the
Denominators of three Factors, in Arith. Progreffion.
98. Examples to the above Series.
101. Notes on Section 7.
102. An Illuſtration of Emerfon's 28th Form for Transformation, in finding the Fluent of
the trinomial Expreffion У ў
1-y+y²
SECTION
VIII.
106. Prop. 13. To find the Sum of a Series when the Numerators confift of two, and the
Denominators of four Factors, in Arith. Progreſſion, being a further
Application of Prop. 12.
109. Examples to the preceding Series.
113. Notes on Section 8.
SECTION IX.
116. Prop. 14. To investigate the Sums of Series, being the Reciprocals of the Powers of
the Natural Numbers.
119. The Quadratures of the different Orders of tranfcendant Hyperbolas exhibited; from
whence is determined the Fluent of the Expreffion
-I
I
&'c.
X x
Sx Si
I
I + x
x.
124. Notes on Section 9.
126. The arithmetical Computations of the Areas of tranfcendant Hyberbolic Curves, by the
Application of Cotes's Formula, in Prop. 7. De Methodo Differentiali.
APPENDIX.
135. Containing General Theorems for 53 different Species or Orders of Series.
187. Examples to the foregoing Series.
212. Promifcuous Series with their Summations, which have been deduced from the General
Theorems; thefe, with the preceding Orders and Examples, exhibiting in the whole
the Sums of near four hundred Series, the greater Part of which are expreſſed in
General Terms.
THE
SUBSCRIBERS to this WORK
Whoſe N A MES have been received, are,
A
THE Rev. Mr. Allen, A. M.
Mr. John Afpland, Soham, Cambridgeshire
Mr. John Ashworth, Manchester
Mr. George Antrobus, Middlewich
Mr. Chriſtopher Afpinall, W. M. and Ac-
countant, Manchester
C
Mr. Chriſtopher Cave, Caifter, Lincolnshire
Mr. John Chesfhyre
The Rev. W. Auguftus Clarke, London,
4 Copies
Mr. John Clarke, Great Ryle, Northum
berland
Mr. Applegarth, W. M. and Accountant, Mr. Peter Clare, Lecturer in Natural Phi-
London
lofophy, Mancheſter
Mr. Thomas Atkinſon, Dunholm, Lin- Mr. Carpenter, Hereford
colnshire
B
Mr. Richard Clegg, Teacher of the Mathe-
matics, Liverpool
Thomas Butterworth Bayley, Efq; F. R. S. Mr. William Cole, Colchester
The Rev. Mr. Brien, Norwich
Mr. John Barnes, W. M. and Accountant,
Dover
Mr. Thomas Barker, Holton, Suffolk
Mr. John Bayley, Middleton, Yorkshire
Mr. John Bartlett
Mr. Bean, W. M. and Accountant, Leeds
Mr. George Beck, Coventry
Mr. Thomas Becket
Mr. Jofeph Bird, Ipſwich
Mr. John Birtles, W. M. and Accountant,
Macclesfield
Mr. James Bottomley, Engraver, Man-
chefter
Mr. Boome, W. M. and Accountant, Putney
Mr. Hugh Byron, Teacher of the Mathe-
matics, Manchefter
George Brockbank, Gent. Cartmell
Mr. John Collier, Rochdale
Mr. William Collier, Salford
Mr. William Coulthard, Officer in the
Excife
Mr. William Crofley, Land Surveyor, Ha-
lifax
The Rev. Mr. Crakelt
Mr. Crooks, Teacher of the Mathematics,
Leeds
Mr. James Cunliffe, Bolton
The Rev. Mr. Clyatt, A. M.
Mr. Thomas Clyde, Litchfield
D
Captain John Daggers
Mr. William Daggers, Kingston, Jamaica
Mr. Richard Dalton, Pool, Carmarthenshire
Mr. Dees, Monkwearmouth, Sunderland
Mr. Benjamin Donne, Lecturer in Philofophy
Mr. Burton, W. M. and Accountant, Li- Mr. James Drurey, Leeds
verpool
George
xvi
SUBSCRIBERS NAME S.
E
George Eaftwood, Efq; London
The Rev. Mr. Eadon, Croydon, Surry
Mr. George Eyre, Castleton
Mr. Mark Elftob, Shotton
F
Mr. Richard Higham, W. M. Accountant,
and Teacher of Geography, Manchester
The Rev. Mr. Holland, Bolton
Mr. James Houghendale
Mr. Hopps, Leeds
Charles Hutton, Efq; F. R. S.
Mr. William Fininley, WV. M. and Ac- The Rev. Mr. Houghton
countant, Thorney
The Rev. Mr. Fiſher, Drax, Yorkſhire
Mr. Fidler, London
Mr. William Francis, Shinfield
Mr. George Francis, Wormhill, Derbyshire
The Rev. Mr. Fofter, Brierley, Yorkſhire
G
Mr. James Gardner, Manchefter
The Rev. Mr. Guilford, London
Mr. William Gibfon, Cartmell
I
Mr. John Jackſon, North Allerton
Mr. William Jones, Hayford
William Johnſon, Efq; Clapham
K
Mr. William Kay, Wakefield
The Rev. Mr. Kenedy, London
Mr. Edward Kendall, Ulverstone
Mr. Kelly, W. M. Bloomsbury, London
Mr. William Green, Land-Surveyor, Man- Mr. William King, Lofthouse
chefter
Mr. John Gilbert, Worfley, 2 Copies
Mr. William Gillmer, Chefter-le-freet
Mr. Graff, Burton
Mr. Thomas Grundy, Clifton
Mr. Zachariah Kirkman, Newton-Lane
Mr. John Knowles, Teacher of the Ma-
thematics, Liverpool
L
Mr. James Lamb, Aldborough, Yorkshire
Dr. Matthew Guthrie, Petersburgh, Ruffia The Rev. Mr. Lawfon, Rector of Swans
H
Mr. John Hadfield, junior, Manchefter
The Rev. Mr. Hammond, M. A. Fellow
of Queen's College, Cambridge
Mr. William Hardy, Cottingham
Mr. John Harrifon, Helsby, Cheshire
Mr. John Harrifon, Prefton
Mr. William Hayes, Frodsham
Mr. William Hampfon
combe, Kent
Mr. Lloyd, W. M. and Accountant, Ken-
nington
Mr. James Loughton, Architect, London
Antonio Mario Lorgna, F. R. S. and Pro-
felor of Mathematics in the Military Col-
lege of Verona
The Rev. Mr. Ludlam, Norton, Leiceſterſh.
M
Mr. Robert Hartley, Teacher of the Ma- John Mathews, Efq; Camberwell
thematics, Daresbury
Mr. Stephen Hartley, Sowerby, Yorkshire
Mr. Jonathan Haworth, Bury, 2 Copies
Mr. John Henfhall, W. M. and Account-
ant, Manchefter
The Rev. Mr. Herbert, Mitcham
Mr. George Hibbert, junior, Manchefter
Mr. John Hill, Merchant, Manchester
Mr. Jonathan Mabbott, Officer in the Exciſe
Mr. John Mathewfon, Tatfield, Durham
Mr. John Medhurst, W. M. and Account-
ant, Manchefter
Mr. Malbourne, Rochdale
Mr. Thomas Malton, Author of the Com-
plete Treatise on Perspective, &c.
Mr. Thomas Molineux, Teacher of the
Mathematics,
SUBSCRIBERS NAME S.
Mathematics, at the Rev. Mr. Ingle's ( Mr. Edward Reed, Weft Alvington
School, Macclesfield
Mr. William Reynolds, Shrewsbury
xvii
Mr. John Morewood, Merchant, Peterſ- Mr. Richard Rideout, Rofs, Herefordshire
burgh, Rusia
Mr. Thomas Mofs
N
Mr. John Nutter, W. M. and Teacher of
the Mathematics, Dewsbury, Yorkshire
Mr. Benſon Norton, Dublin
O
Mr. Hugh Oldham, Steward to the Right
Hon. Lord Ducie, Manchefter
Mr. Adam Oliver, Worcester
Mr. Alexander Rowe, Reginnis, Cornwall
Mr. Ifaac Rowbotham, Wefthaltam
The Rev. John Rofs, Portsmouth
S
Mr. George Sanderfon, London
Roger Sedgwick, Efq; Mancheſter
Mr. Thomas Simpfon, Glerk to the Navi-
gation, Halifax
Mr. John Smith, W. M. and Teacher of
the Mathematics, Warrington
Mr. John Orrell, Teacher of the Mathe- Mr. George Stevenfon, Hazen
matics, Bolton
The Rev. Mr. Ofwald, M. A. London
P
Mr. William Sherwin, Afton-upon-Trent
Mr. James Spencer, Merchant, Salford
Mr. John Shadgett, Officer in the Excife
Mr. Paul Palmer, Mafter of the Academy The Rev. Mr. Shawe, London
at Mitcham
Mr. John Pearfon, Lee-fair Green, Yorksh.
Mr. William Penn, Chalfont, Bucks
Mr. William Perry, Kelfo
The Rev. Mr. Pearce, Deal
Mr. George Perrott, Hefle, Yorkſhire
Mr. Lee Phillips, Mancheſter
Mr. Robert Phillips, St. Agnes, Cornwall
The Rev. Mr. Pope, Stand, Lancashire
The Rev. Mr. Popple, M. A. of Trinity
College, Cambridge
Mr. Thomas Porter, Salford
Mr. Powle, Teacher of the Mathematics,
Hereford
Mr. Robert Pulman, Mafter of the Aca-
demy in Halifax, Yorkshire, 6 Copies
R
Mr. Richard Rawle, Redruth
Mr Ifaac Slee, Plumpton, Cornwall
Mr. Suddones, WV. M. & Accountant, Chefter
Mr. Jofeph Scott, Cawthorne
Captain Abraham Scott
T
Mr. Samuel Taylor, Mancheſter
Mr. Ralph Taylor, Teacher of the Mathe-
matics, Stretford
Mr. Michael Taylor, Marley Hill
M. Antonio Taurant, La Baje Ville, Calais
Mr. Ralph Thompſon, Witherley Br.
Mr. John Thompſon, Northallerton
Mr. Thomas Truffwell
Mr. John Travis, Manchester
W
Mr. Ward, Architect, St. Mary Gray, Kent
Mr. Walker, Mafter of the Academy, at
Leeds, Yorkshire
Mr. Rigge, Teacher of the Mathematics and Mr. Daniel Walker, late W. M. and Teacher
Land-Surveyor, Cambridge
of the Mathematics, Manchester
Mr. John Rivett, East Dereham, Norfolk | Mr. Richard Walker, Whitefield
Mr. John Ryley, Northampton
Captain Robert Ravald
Mr. Ifaac Rawlinfon, Cartmell
Mr. William Watfon, Alnwick
Mr. John Wallace, Dover
Mr. William Wefton, Chester
Mr. James
xviii
SUBSCRIBERS NAME S.
Mr. James Whalley, Teacher of the Ma- | Mr. William Winn, Thirsk, Yorkshire.
thematics, Bolton
The Rev. Mr. Wildbore
Charles White, Efq; F. R. S. Manchefter
The Rev. Mr. Whitaker, Rector of Ruan,
Cornwall
Mr. William Wilfon, W. M. and Account-
ant, Stockport
Mr. Willan, Teacher of the Mathematics
Mr. John Willis, Marſk
Mr. Philip Wright, Stafford
Mr. James Whitaker, Attorney at Law, Mr. Thomas Woolfton, Sulgrave
Salford
Mr. Daniel Whitaker, fenior, Manchefter
Mr. Henry Whitaker, W. M. and Teacher
of the Mathematics, Manchefter
Mr. Stephen Williams, Truro, Cornwall
The Rev. Mr. Wilfon, A. B. Vicar of
Whitchurch, Yorkshire
The Rev. Owen Wylde, Canterbury
Edward Wychcott, Efq; Hammerfmitth
Y
Mr. Jofeph Yates, Manchefter
The Rev. Mr. Young, London
Mr. James Young, Teacher of the Mathe-
marics.
SUBSCRIBERS NAMES Continued.
A
R. Jeremiah Ainſworth, Master of
MR
the Mathematical School, Mancheſter.
The Rev. Mr. Atterfley, London.
Mr. Robert Abbatt, jun. Prefton
Mr. John Aikin, Norwich
Mr. John Arnold, London
B
Mr. Richard Batho, Tilflock, Shropshire
Mr. Thomas Barrow, Welton, Yorkſhire
Mr. John Barrow, Manchester
Mr. Hugh Barnard, Stoke Biſhop, near
Briftol
Mr. Edward Boucher, Taunton
Mr. John Bonnycaftle, London
Mr. John Buckley, Rochdale
Lieutenant Miles Bowers
Mr. Bonner, Halnaby
C
Mr. D. Chapman, Woodhouse, Leiceſterſh.
Mr. Robert Cargill, Stoke Bishop
Mr. Ifaac Clarke, Boskfeller, Mancheſter
Mr. John Clarke, Lincoln
F
The Rev. John Fawcett, Mafter of the
Boarding School, at Brierley-hall, in Midg-
ley, near Halifax
Mr. Samuel Falkner, Bookfeller, Mancheſter
Mr. Thomas Fawkner, W. M. and Account-
ant, Manchester
Mr. Jonathan France, Master of Hope
School, Derbyshire
Mr. John Fatherley
Mr. John Fletcher
G
Mr. Gadbury, Covent-Garden, London
Mr. John Grime, fen. Collyburſt, Manchester
Mr. Richard Green, Manchester
H
Samuel Horfley, L. L. D. S. R. S.
The Rev. Mr. Urbane Hallam, London
Mr. John Hampſhire
Mr. R. Hoofsetter, London
Mr. John Haflingden, Bookfeller, Manchefter
Mr. William Hedley, Cambo, Northum-
berland
Mr. James Cook, Attorney at Law, Salford Mr. William Hodfhon, Northallerton
D
Mr. Jofeph Denton, Holymore
Mr. John Duncombe, Profeſor of Mathe-
matics and Philofophy, Liverpool
E
Mr. John Eadon, jun.
Mr. Elliott, Mytham-Bridge
Mr. John Hamfon, Helsby
Mr. William Hough, IV. M. and Account-
ant, Norwich
I
Charles Ingham, Efq; Ilington
Mr. John Jackſon, Land-Surveyor
Mr. Jofeph James, Stoke Bishop
1
<
The
XX
SUBSCRIBERS NAMES Continued.
K
The Rev. Mr. Kenyon, Salford
L
Mr. James Lamb, Aldbrough
M
The Rev. Nevil Maſkelyne, D.D. F.R.S.
and Aftronomer Royal
George Matlicws, Efq; London
Mr. William Marfden, Netherhurst
Mr. John Meaſures, Barrow-upon-Soar
Mr. William Macaulay, Manchefter
N
Mr. Edward Nairne, F. R. S
Mr. Dan Newton, Mafter of the Board-
ing School, at Groppenhall
Mr. John Needham, Sheephead, Leicester fh.
Mr. John Norman, Braybrook, Northamp
tonſhire
P
Mr. John Parker, Ashby de la Zouch
Mr. Thomas Peat, Teacher of the Mathe-
matics, in Nottingham
Mr. John Peck, Segglefthorpe
R
Mr Roberts, Teacher of the Mathematics
S
Mr. Savage, Stafford
Mr. John Seddon, Salford
Mr. James Sleech, Derby
Mr. J. Smith, Hetherfett, Norfolk
Mr. Stuchfield, Stepney
Mr. William Swift, Stow, Lincolnshire
T
Mr. Francis Taylor, Bridlington
Mr. Richard Todd, Alnwick
The Rev. Mr. Turner, of Magdalen Hall,
Oxford
Mr. Francis Turner, Lechlade, Glouceſterſh.
Mr. John Turner, Witney
Mr. Richard Thornhill, Mancheſter
V
The Rev. Thomas Vaughan, M. A.
Morpeth
Mr. Charles Vyfe, Author of the Tutor's
Guide, &c.
W
Mr. William Wales, F. R. S. Mafter of
the Royal Mathematical School, in Chrift's
Hoſpital, London
The Rev. Mr. Wallworth, Putney
Mr. William Ward, Leicefter
Mr. Thomas Watkins, Bristol
Mr. Rowland Wetherald, Bishop-Wear-
mouth
Mr. John Woolfenden, W. M. and Ac-
countant, Manchester
Mr. Thomas Wood, Stoke Golding
Mr. Peter Warfdale, Stockton-upon-Tees
Mr. William Walton, Newcastle
Mr. John Wright, Aldbrough
[ x ]
A
DISSERTATION
ON
THE SUMMATION OF INFINITE CONVERGING SERIES
WITH ALGEBRAIC DIVISORS.
"B
terins.
SECTION I
DE F. I.
Y the Sum of an Infinite Converging Series, is underſtood a, Formula expreffing
the aggregate or total value of a feries confifting of an infinite number of
D. E F
II.
2. By the Summatrix of a Series, is underſtood an Expreffion for the fum of an in
definite number of terms of that feries; and which has been uſually called The General
Sum of the Series for an indeterminate Number of Terms.
PROP.
3. To find the Sum of a Series terminating in
ber of Terms.
L.
I
; wherez denotes the Num-
p + q x
ċ
For
2
2
THE SUMMATION
For the indefinite quantity z let the feries of natural numbers from unity be pro-
greffively fubftituted, and there will arife the feries
I
I
I
+
+
p + q
p+29
p+39
I
+
+ &c.
p +49
I
p + qz
Now let each term of this ſeries be multiplied into the correfponding one of the
xp:q+¹, xp: 9+², xp:9+3, xp:9+4, &c.
feries
}
and they will become
xp:q+1 xp:9+2 xp:9+3 xp:9+4
2+29 p+39 P+49
p + q
+
+
+
+ &c.
Now by putting S for the fum of this ſeries, and taking the fluxion, there ariſeth
q
S
= *p:9 * + *p;q+1 * + xp:9+2 ¿ + &c.
Divide this equation by xp:9
and it gives
9
I
= 1 + x + x² + 23 + 2.4. + &c. =
x p : q x
I
x
x p : 9 x
therefore $ =
and the affumed fluent S =
"
q × I-x
HS A
'x P : 9 x
I-X
xp:9+1
+
p+q
xp:9+2
xp:9+3
+
p.+ 29. p. +39
+ &c.
Confequently, by making
I in the finite quantities, we have
expreffing the fum of the infinite feries
I
I
I
I
(A) + +
+
+
+ &c.
p+29 p+39 +49
I-X
provided the fluent be fo taken, that it may, vaniſh when x. o, but perfectly
integral when x = 1. And we ſhall here obſerve, once for all, that whenever
an expreffion for the fum of a ſeries is inveftigated, the fluent is always understood to
be taken ſuch, that it may vaniſh-by making x = 0, but may become perfectly inte
gral by making x = 1..
1
PR. O P.. II.
4. To determine the Summatrix of the preceding Series.
Becauſe the fum of the feries is expreffed by:
S
'xp: q x
•
1
and
OF
3
CONVERGING
SERIES.
and z denotes the number of terms, for p wrire p + qz, that the fluent may become
I
S
'x p + q≈:?
then will
- | ས
9
S
be the required fummatrix.
1
xp: q x
I-X
1
9
S
xp+qx: 9 x
25:
I -
For having thrown the quantity
multiplied it by x, and taken the fluent, it becomes
p+qxx + 2 p + q x z +3
1, arifeth the feries
xp: q+x+I
xp:9+x+2 *p:9+x+3
+
+
p + q x z +1
from whence by making x =
I
(B)
+
I
•
+
I
xp + q x: 9
into a feries,
X-I
+ &c.
I
+
+ &c. .
p+q⋅ x+4
p+q.z +1
2+1. p + q z+2 p+q·x+3
Z
Now, for z put any whole number at pleaſure, and collate the refulting feries with
the preceding one A, and the initial term of the feries B will evidently be the z+1th
term, or the next ſubſequent one to the th term, of the ſeries A; and therefore the
ſeries B will be the feries A leffened by its z initial terms. Confequently the feries.
B being taken from the feries A, the remainder is the fum of as many of the initial :
terms of the fame feries A as there are units in z. But
and
I
- S
I
9
S
xp: 9
=A, by Art. 3.
x p + q z: 9 x
=.B;
I - X
therefore
; S
xp: q x
I
X
S
x p + q x : 9 x
13
9
I. x
is the fummatrix of the feries A for z terms. Q. E. I. & D.
I
p + q
PROP.
III.
5. To find the Sum of the Series
p+29
By proceeding as in Art. 3.
XP: L X
I
+
I
I
&c. ±
p+xq
+
$+39. p+48 p+59
the following equation is derived,
*³ + *
&c. =
+ *
hence
4
THE SUMMATION
hence Ś
11
x P : q x
and the fluent S =
;
q x
1 + x
xp:9+1
xp:9+2
xp:9+3
+
9
S
xp: 9x
I+x
xp:9+4
+ &c.
P+49
p+29 P+39
= 1 in the finite quantities, becomes
p + q
which, by making
I
I
I
I
(C)
p + q
+
+ &c. Q. E. I.
1+29
p+39
1 +49
PROP.
IV.
6. To determine the Summatrix of the Series, in the last Propofition.
For p, in the expreffion for the fum of the feries
it becomes
9
S
x P : q x
1 + x
• put p+28%, and
I
S
9
x p + zqx: 9 x
;
I+x
xp:9+2x+1
which being reduced into a feries, and the fluent taken, gives
xp:9+2x+3
xp:9+2x+2
+
&c.
p + q. 2 x + I
p + q. 2x + 2
P+9=2x+3
2%
And if x be made = 1, the feries arifing will be
I
(D)
p + q. 2 x + I
I
+
I
P+q.. 22+2 P+ 9.22+ 3
&c.
Having fubftituted for z any whole number at pleaſure, compare as before the ſeries
C and D, and the feries D will appear to coincide with, and begin from, the 2x+1¹
term, or the next ſubſequent to the 2 th term of the feries C. It is therefore mani-
feft that the ſeries D is equal to the feries C leffened by the 2x initial terms of the
fame feries C. Confequently if we take D from C, the remainder will exhibit the
fum of as many terms of the feries C as there are units in 2%. But
I
C:
C =
S
xp: 9 x
I
and D =
>
9
1 + x
9
I
S
xp: 9x
q
I + *
is the fummatrix of the given feries. Q E. I. & D.
9
S
x p + 2 9 : 9 x
[ + *
x p + 29x : 9 &
; therefore
Į + x
PROP.
OF CONVERGING SERIE S.
5
PROP.
7. To find the Sum of the Series
V.
'I
★ + q x m
I
+
p + 29 × m
I
P + 3 9 × m³
+1
I
+ &c.
p + 49 × m
in which m is any affirmative Number affumed at Pleasure not less than Unity.
Let each term of the feries
xp: q+1, xp: 9+2,
p:9+3, xpq+4, &c.
be fingly multiplied into the correfponding terms of the given feries, and there will
arife
xp:9+1
+
xp:9+2
p + q m p+29 m²
•
+
xP:9+3
m³
+39
xp:9+4
+
+ Sic.
p + 49 •'m+
The fum of this feries being put S, the fluxion taken, and divided by xp:9 x,
becomes
it
I
I
x P : 9 x
m
m²
+
-12
I
It
+ &c.
3
m
m
But the fum of this feries is known to be equal
I
S.
; therefore
m = x
and the fluent S =
(by making x = 1 as before)
m ± x
S
xp: 9x
xp: 9+1
-
+
9
+
p + q
m
xp:9+3
p+29
xp: 9x
m²
+ &c. =
I
=
m ± x
I
I
+
p+q. m
p+29. m²
I
I
+1
+ &c. Q. E. I.
p+4q.m+
p+ 39. m³
PROP. VI.
8. To find the Summatrix of the preceding Series.
Becauſe the fummatrix of the feries
།
+
m
ន
m
*
method we have juft now fhewn, to be
m -
२
x
+
&c. to be
m
m³
m+x
+
m
+ &c. is found, by the
;
and that of the feries
m
x2x+1
in the former of
2 z
m²² × m+x
which as many terms are fummed as there are units in z, and in the latter as many as
there are units in 2x; therefore let both the feries and the formulæ for their fumma-
trices
6
SUMMATION
THE
trices be multiplied by xp: x, and the former will become refpectively
xp: 9 x
:9
xp: q+1 x
+
xP:9+2x
m
m²
+
and the latter
23
± &c.
z
mz
772
X
and
M² x — x 2 x x x P : 9 x
m² z.m+x
×
Taking therefore the fluents,
M 2 xx X xp: 9 x
*
and making ≈ = 1 in the finite quantities, the ſeries arifing will be
I
+
I
+
I
p+q.m p+29 m² p+39. m³
the fummatrix of which when the figns are conftant is
I
S
9
1
[
+
+ &c.
p+49. m².
mz x x x xp: 9 x
MX
m
•
but when they are alternately changeable from pofitive to negative,
I
9
M22
- 2x x x P : q x
m² x . m + x
}
provided that in thefe alfo, they become perfectly integral by making » = 1.
Q. E. I.
:
•
NOTES,
OF CONVERGING SERIES.
7
NOTE S,
EXPLANATORY AND CRITICAL,
ON SECTION I.
DEF. I.DY the infinity of a Series is meant the infinity of the number of its
B terms. But as a number actually infinite is a manifeft abfurdity to
our ideas of numbers in general, fince we cannot poffibly conceive them to exift ab-
ftractedly from fomething limited and determinate, this term has given occafion to as
many various difputes, as there have been various opinions concerning it; which,
however, we ſhall here paſs by, as being quite foreign to our purpoſe. For though
we cannot abfolutely fay in things which have a real exiſtence, that their number is
greater than any affignable number, or that all their units can be actually expreffed,
when continued without limitation; yet this must be granted by every one, being a
felf-evident truth,that we can conceive a number greater
than any
determinate
number whatever, and after that a greater, and fo on without end. And confe-
quently, whether it be poffible or not, for the number of any things really exifting
to exceed any affignable number, nothing can be more obvious, than in fucceffive
productions, they may become greater in number than any given number whatever ;
becauſe, though at any given term they may be faid to be finite, yet they may be
increaſed without end.
From hence we have a diftinct notion of the fum or total value of an Infinite Series.
For nothing would be more abfurd than to imagine it a collection of ſeveral particular
numbers that are connected and added together one after another, as this would fup-
pofe the parts of an Infinite Series to be all known and determined, which we have
juſt obſerved cannot all be ſeparately affigned, there being no end in the numeration
of its parts. We must therefore conceive it to be poſitive and determinate fo far as
the ſeries is actually carried on, with an endleſs addition of its terms that can be fuc-
ceffively made to it; and confequently in this fenfe it may be faid to have a fum, or
total value, whether it can be exprelled by any means or not. . Now in any Infinite.
D
Series
8
THE SUMMATION
Series of which the terms grow lefs and lefs, or converge to a point as it were, the
point of convergency, or rather the ſpace included by the converging lines, is the
limit which the whole feries, taken in its infinite capacity, cannot poffibly go beyond
or exceed; and therefore there may be a number affigned which the fum of no given
number of terms of the feries can ever reach, nor indeed ever be equal to; yet it may
come fo near it as to want lefs than any affignable difference. Here then is the precife
idea of the fummation or total value of a feries confifting of an infinite number of
terms, as expreſſed in the definition; which is evidently not taken as a number found
by the common method of addition, but as fuch a limitation of the value of the ſeries,
that if all the terms could poffibly be added together, the fum would be equal to this.
It is hardly neceffary to obferve, that if the feries do not converge, that is, if each
fucceffive term le not less than the preceding one, it can have no limitation or total
value; and therefore as no value can be affigned for an infinite feries in this cafe, it
may be faid to have no determinate fum in any fenfe whatever; from whence the fum
of ſuch a ſeries is ufually faid to be infinitely great.
£
PROP. I. Here our Author lays down the fundamental propofition, or ground,
work, upon which his whole ftructure is built. This ought therefore to be particu-
larly attended to, and well underſtood by the young Mathematician, before he pro-
ceeds any farther. And to give him all the affiſtance he can wish for, not only in
this, but through the whole treatife, we fhall endeavour to be as explicit as poffible
in our remarks, and by deſcending to the very minutia of each propofition, leave
nothing in obfcurity, or any ways myfterious. For it is certainly the moſt eligible
method of treating any fcience, either in the primary capacity of an Author, or in
the fecondary one of commenting on other's performances, to render it as clear and
intelligible as the fubject will admit of; for where it is, otherwife, it must be done,
either intentionally, or (which I believe is more commonly the cafe) through want,
of a clear perception in the writer. The former is, the moft ridiculous pedantry, and,
the latter almoſt unpardonable. We have, however, feveral inftances of this kind
in the Comments and Explanations of the writings of our moft celebrated authors in,
the mathematical way. It muft indeed be acknowledged, that an author, may often-
times, without the imputation of pedantry, adapt his manner of treating a ſubject to
the capacities of thoſe only who are already adepts therein, and which muft of courſe.
appear obfcure to thoſe who have but fuperficially adverted to it; but when notes and
o fervations are given to fuch a treatiſe profeſſedly explanatory, the leaſt ſhade of ob-
ſcurity in theſe is as prepofterous as falfe concord would be in a treatise on Grammar,
For this reaſon I have all along fhewn not only how the fluxion and fluent of each ex-
preffion are deduced, but alſo have given the process at length of drawing the infinite
(eries from the fractions, transforming the quantities, and fuch like. This method
I have
I
<
OF CONVERGING SERIES.
9
I have purſued in each propofition; unleſs when an expreffion of the ſame kind oc-
curs, where it would have been quite fuperfluous.
We ſhall begin with inveſtigating the fluxion of the feries
xt: 9+1
p+q
xp: 9+2
+
+ &c.
p+29
By proceeding in the common method we have
P: 9 + 1 x x P : 9 x
+
p + q
but p:9+1 =
p + q
+9
, p:9 + 2 =
Þ : 9 + 2 × xp: qts &
p+29
p+24, &c.
+ &c.
9
q
1+9
I p+29
hence
p + q =
:p+29 = &c.
9
9
9
xP: 9x
confeq.
+
+ &c. =
9
9
and xp: q‡ + xp : q† iż + &c. = 48.
And
I
IX
is thrown into a feries thus,
Ì—†) Í
I
+x
+
+x
+
+
(I + * + *² + x³ + &c.
&c.
And that the fluential expreffion for the fum of the feries A is truly affigned, may
be eaſily aſcertained by actually drawing out the fluent, as follows,
P
I−x) x 9 x
P
+2
? * + * ?
* + * i
* + &c.
P
* I *
x 9
+ * 9
+ * 9
+
x -I
P
+ x q
+ £ ?
D 2
+
&c.
and
1Ο
THE SUMMATION
and by taking the fluent of each term, we have
P
2 +1
x 9
P
x 9
Р
+2
+3
x q
+
+
+ &c.
P
Р
Р
+1
+2
+3.
9
9
9
xp:9+1
which multiplied by
becomes
+
p + q
xp:9+2
P+29
+ &c. the affumed feries.
P
Since other forms of fluents for the expreffion
x 1 x
q. X I X
= 1
may be deduced than the
above, it is a neceffary reftriction that it be taken fuch that the whole may vaniſh
when xo; otherwife the fluent would need a correction, whereby the numerators
of the terms might become fractional numbers; which would be wholly incompatible
with the method of fummation here propofed, where the numerators in the terms of
the fluent muſt neceffarily be integers. And in this fenfe it is always to be under ftood
when the fluent is faid to be perfectly integral, the flowing quantity not affecting the
denominators. It may not be improper juſt to obſerve likewiſe, that it might perhaps
be ſuppoſed by the young reader, that if x be taken in the fluent, it might alſo
be taken the fame in the fluential expreffion; which would therefore become incon-
fiftent. To reconcile which he muſt remember, that it is abfurd to affume any de-
terminate value for while in a flowing ftate, as it evidently must be under the
fluential fign; for if it were conftant it could have no augment or fluxion, and
confequently there could be no relation between cotemporary increments, fince it
would then be x = 0; and therefore the quantity under the fluential fign will vaniſh,
as it ought to do. For it is obvious there can be no fluxion where there is no flowing
quantity to generate it.
PROP. II. The quantity
1−x) xp+qx : q
x p + q z: q
may
I-x'
་ ་
be refolved into a feries as follows,
(xP+qx: 9 + xp+q.x+1:9 + xp+q·x+2:9 + &.£..
x p + q ~ : 9
مه
-xp+qx: 9 x
+xp+q.≈+1:9
+ xp+q.x+1:9
+ xp+q.x+2:9
xp+q.x+1:9 x
&c.
+ &c.
xp+qx:q+1
And the fluent of this feries (divided by q) is derived in this manner,.
`x p + q 2 +1:7+1
+
+ &c.
p+qx:q+1x q
p+q·x+1:9+1 x 9.
but
+
OF CONVERGING SERIES.
II
but p+qz9+1=
44
+x+1, p +9.2+1:9+1=
p + qz + q
9
P
+1=
q
+≈ +2, &c.
Therefore theſe quantities being each multiplied by q become reſpectively
2
Z
1 × p: q + x + 1 = p + q x z + 1, q x p: 4 + x + 2 = p + q x z +2, &c.
confeq.
xp:9+x+1
•
+
xp:9+x+2
p+9.x +2
+ &c. as in the propofition.
Now if for z we put any whole number, fuppofe 3, then the feries B will
become
I
I
I
+ &c.
+
+
P+49 P +59 p+69
which is evidently the feries A commencing at the 4th term thereof; and therefore
the terms of B muſt coincide with thoſe of A from the 4th term, when both are con-
tinued in infinitum. From whence nothing can be more obvious, than when the in-
finite ſeries B is taken from the infinite feries A, the remainder will be equivalent to,
or truly exprefs, a finite number of initial terms (z) of the feries A, which in this
cafe will be
3 p² + 12 p q + 119
I
I
I
+
p + q p+29
+
or
P+39
p² + 6 р`q + x 1 p q² + 6 q ³°
I
PROP. IV. The fluential expreffion may be expanded thus,
1+x) xp+2qz:q x
(xp+2qz:9x -=1+292:9+ s + Sic.
xp+2qz:9 x + xp+29≈:9+1 &
xp+29<:9+1 ¿
xp+29≈: q+1 i
Hence the fluent
+ &c.
xp+2qx:9+
xp+29≈:9+2
I
+ &c. being multiplied by
p+29x:9+1
+29:9+2
q
becomes
xp:9+2x+I
xp:9+2x+2
+&c. as in the propofition.
p+q.2x+ı p+9.2x+2
Let zany whole number at pleaſure, fuppofe 2, then will the feries D become,
I
P+59
I
I
+
p+69 p+ i q
&c.
which is manifeftly the continuation of the feries C, from the 22+1th or 5th term
thereof. Confequently if D be taken from C, both confidered in their infinite capa-
city
12
THE SUMMATION
city, there will remain the finite value of 2z initial terms of the feries C; which in
this cafe will be
I
p + q
I
Σ
I
+
1+29 p+39 +49
PROP. V. The fluxion of the feries multiplied by q is
xp: q x
+
m
xp:q+1 x
m²
xp:9+2 x
2
+
M3
± &c.
may be reduced thus,
x
± +
m
2
9
11
I
And the fraction
mx
mx) I
I F
m
x4
It
+1
ន
m
x
m
m²
+
± &c.
m³
+
m²
+廾
​++
3
m
m
&c.
And that the fluential expreffion is truly affigned, may be eafily determined as
+1
follows,
x p : q x
m = x) xp: q x
m
xpiqti x
xP:1 x =
m
}
:
xp+q + *·
m²
+
xp:q+2 x
m³
+ &c.
xpiqti
+
m
xp: 9+1 x
xp: q+2 x
m
m²
+
+
xp:9+ 2 x
m²
xp:9+2 x
+ &c.
m²
Hence
OF
13
CONVERGING SERIES.
Xp:9+1
xp:9+2
Hence
xp:9+3
+
+
+ &c.
p: 9 + 1 x m P:q+2xm²
P: 9+ 3x m³
3
which divided by q becomes
x1:9+1
p+q.m
xp:q+z
+
+ &c.
p+24.m²
PROP. VI. The fummatrix of the feries
duced,
x
+ + &c. may be thus de-
Since is the fum of the given feries, it is plain that if z denote any number
m-x
x
|!
x
m
of terms thereof,
x z+1
mxxx
will be the next fubfequent term,
xx+2
mx + 2
the next, and fo
on.
But this feries continued ad infinitum is =
xx+1
mx.m-x
-5
which appears by ex--
panding the expreffion,.
mzti
x+
-
m² x) xx+1
+
xx+2 xx+3
+
mx+z m²+3
+ Sic..
xx+z
m
+
xx+z
m
xz+2
:
xx+3-
+
ន
m
m²
xx+3
+
m².
xx+3
+
&c.
m²
+ &c.
And after the fame manner may the fummatrix of the feries be difcovered, when the
figns change alternately, from pofitive to negative. For
m+x
is the whole fum
of the feries in this cafe; whence if 2 x denote any even number of terms thereof, z
x 2x+1
x2x+z
m ²z+1² m² z+2
will expreſs the pairs of terms contained therein, and conſequently
&c. will be the terms fucceeding the 22th term; the firft of which will evidently be
affirmative whatever number z be expounded by. But the fum of this infinite feries-
when the figns are + and
alternately is
x2z+1
m²z.m+x
; which is alfo. manifeft from
throwing
14
THE SUMMATION
throwing the expreffion into a feries,
m 2x+1 + m 2x x) x2x+1
* 2x+1
m²zti
x 2x+2
m²x+z
+
x 2x+3
m2x+3
&c.
xzx+2 m² Z
x2x+1 +
1
The fluent of the ſeries
m 2x+1
x 2x+z
m
x2x+2
x2z+3 m2xZ
77L
m²x+2
x 2x+3
+
m²
+
x2x+3
m²
2
+ &c.
&c.
xp: q
xp: q+1 x
+
#1
m
7/22
+ &c.
is eaſily deduced thus,
xp:q+1
piq+1.m
which drawn into
I becomes
q
xP:9+1
p+q.m
+1
xp:9+2
+
+ &c.
p:9+2.m²
xp:9+2
+ &c.
p+29.m²
1
1
SECTION II
OF CONVERGING SERIES.
15
SECTION II.
PROP. VII.
9. To find the Sum of a Series, when the Numerators of the Terms are common, or confti-
`tute an equal Series, and the Denominators confift of any Number of Simple Factors; the
General Term being
(E)
p+qz.m+nz.r+sz.t+uz. &c.
where is the index of the terms.
From Art.
3. and 5. we have
I
F:
sxp: 9 x
9x
xp:9+1
±
q
p + q
IFX
*
this equation being multiplied by x; a—p; q -x, and the fluent taken, becomes
xp:9+2
2+29
xp:9+3
+
+ &c.
P+39
I
G =
n
SF
xm: n+1
**
m: n—p:
Fx min-p÷q − 1 x =
+
p+q x m+n
xminti
p+29 Xm+2n
+
xm:x+3
+ &c.
p+39 × m + 3n
This new equation drawn into : —— and the fluent taken gives
xris
Gxr:s—m: n−1 ; ——
12 -1
H = ÷ S
I
SG
xris+3
+
p+39 × m+31 × r+35
:-
ૉ'
aristi
S
p + q x m + n xr+s P+ 29 X m + 2n xr+ 25
+ &c.
·
This equation being again multiplied by x, and the fluents of the
refulting expreffions found, there arifeth
I=
I
น
→ SI ✰
x114+1
17
14
+
p + q Xm+" x r+s X i+u
+ &c.
xx:4+3
p+ 29 Xm+2n X r+25 x 1+244 k#39 × m±3″ × r+35 × ¿+3u
and ſo on ad infinitum, the law of the progreffion being evident. If therefore in the
fingle terms of the feries, as alfo in thofe of the fum (abferving to correct the fluents,
E
if
16
THE SUMMATION
ولها
if neceffary) we make x = 1, and multiply both members of the equation by the
refult will be the fum of the feries (for any given number of factors) of which the
general expreffion is
3
p+q% × m+nz × r+sz × t+uz &c.
10. By the fame method may the fummatrices of thefe feries be derived. For
fince
I
s
xx x xp: 9 x
I-X
9
is the Summatrix of the Series A (Art. 4.) when the figns are affirmative; and
I
s
9
I 1-x2≈ x x P : 9 x
I + x
the fummatrix of the feries;
t
when the figns change alternately (Art. 6.) we have only to fubftitute the refpec-
tive fummatrix in the place of the fum F in the preceding propofition, and proceed
exactly in the fame manner; which will be occafionally fhewn hereafter.
11. Before we proceed to the application of this method, we fhall premiſe the
following
LEM M A.
*fy
The fluent of the expreffion contained under two fluential figns, may
always be obtained either algebraically, or by algebraic quadratures, provided y be
expounded by x, and by x.
For let the fluent of be taken, and prefixed to the firſt interior fign, that it may
become ≈ fy*; let z be alſo multiplied into yx, the fluential fign prefixed, and ſub-
tracted from the former quantity; then will
Z
*/*-/yx = //
x
This is evident by taking the fluxions on both fides the equation, for then it
becomes
żƒÿx+yz ż−yzż = żſy*,
where both fides of the equation are manifeftly the fame; and confequently the
fluents of x/yx and yz* will expreſs the whole fluent of Лy the given ex-
z
preffion; which fluents may therefore be had either purely algebraically, or by the
algebraic quadratures.
12. And generally may any compound fluential as
&c. S i S ù S * S ÿ ÿ
yx
be
}
OF CONVERGING SERIE S.
1༡
that
be refolved into 2, 4, 8, 16, &c. fimple fluentials, by proceeding in the fame man-
ner, provided y be expounded by x, and ż, ù, i, &c. be fo expreffed by x, x,
the fluents of ż, ù, †, &c. uż, tù, ż ſtú, may be expreffed algebraically. This
is obvious; for the interior fluential involving two figns, may be refolved into two
fimple fluentials by the Lemma; and thefe being each multiplied by i, and the figns
prefixed, there will arife two fluentials each involving two figns, which may there-
fore be refolved into four fimple fluential quantities; and fo on at pleaſure.
13. We fhall now preceed to illuftrate this Method, by confidering thofe feries the
Denominators of which confift of two fimple factors; beginning with thofe which
admit of a pure algebraic expreffion for their fum, and which may be generally re-
prefented in this form,
I
I
I
+
p+39.m+3n
p+q. m + n p+29.m+2n
I
p+qz. m +nz
The general formula for the fum of this feries is
I
G = 1/1/1 / F x
I
H
+ &c.
p +49. m +4n
Fxm:n p:9-1 (Art. 9.) or by fubftituting the value of F
I
S
xm : n—p: q—1 x
q n
1 x
Divide this compound fluential into two fimple ones by Art. 11. and it becomes
S
x P : 9 x
xm:n―p: q
xp: q x
I
m
ng X
n
P
9
IF X
m
P
nq X
S
IF X
72
9
and make x = 1 in that part of the expreffion without the fluential fign, ſo that the
whole formula may become
I
xp: 9
m
nq X
12
P
૧
* *: * X *
IF*
which is the fum of the propofed infinite feries, expreffed in general terms; and which
for the fake of diftinction we fhall hereafter call the Ecumenical Formula.-When
this is applied to any particular cafe, as to ſeries of the above form of which the figns
are affirmative, we muft obferve, that if
be a whole number, the fum of
the given feries may be expreffed purely algebraically; for in this cafe the cecumenical
m
22
9
formula becomes (putting
m
n
P
E 2
I
ng o
18
THE SUMMATION
π
I
and this is =
n qπ
S
12
I
q
xπ X xp : 9 3
I
**
1+x+x²+x³+x++ &c. × xP:9; the fuent
I
of which is
X:
nπ
xp: 9+1
P+9
xp:9+2 xp:9+3
+
+
+ &c.
p+29
p+39
xp: q+r
p+q
If therefore we put
= 1 as before, it will be
1
I
I
I
I
(L)
X:
n π
+
p + q p+29
+
+ &c.
p+39
p + = q
T
the fum of the propofed feries, the general term being
where as
many of the initial terms will be collected as there are units in . But if the given
ſeries be affected with affirmative and negative ſigns alternately, the Algebraic Sum
cannot be obtained unleſs be an even whole number; for then the oecumenical for-
mula becomes
I
p + qz x
m + n z
I
S
-xπ X xp : q x
I + *
the terms of which will be perfectly integral when expanded by making ≈ = 1.
Hence
I
い
​X:
71 π
xp:9+1
p + q
xp:9+2 xp:9+3
p+29 P+39
xp: q+x
+
&c.
p+ # q
I
I
I
I
= (by making = 1) (M)
X
X:
+
&c.
nπ
p + q
p+29
p+39
I the fum of the propoſed ſeries when the figns change alternately.
p+9
14. By a fimilar method to that fhewn in Art. 19. may the fummatrices of theſe
feries be obtained. For in the formula G, Art. 9. let there be fubftituted the value
I
of the fummatrix of the firft feries F (-
9
S
I X ≈ X xp : 9 x
-] ----.
-1-*
Art. 10.) when
G=
Г.
xm:n−p: q−x x
Ї
the terms are affected with pofitive figns, and there arifes
I
ng
I-** X *p: ż
I-X
If this fluential be divided into two parts, and the refulting expreffions converted into
m
Р
one, making
≈≈, andx=1-in the quantity without the fluential fign,
72
9
the integral will be expreffed in this moft fimple form
I
пят
SE
I -xπ
Xπ X I
xx XxP: 9
1-*
Which
OF
19
CONVERGING SERIES.
Which being expanded, the fluent of each term taken, and afterwards ≈ be expounded
by 1, we get
T
92
(N) 12 x :
ηπ
I
I
p + q x p + q . z+1
I
+ =
+ &c.
p +29 xp +9.≈+2
the fummatrix required;
where as many terms of this feries
p +πq xp + q.x + x
muſt be taken as there are units in, and x (being any whole number taken at plea-
ſure) is the index of the number of terms in the propofed feries which are fummed
thereby.
15. And by a like procefs may the fummatrix of the fame feries be had, when the
figns are changeable; provided that be an even whole number. For in this cafe the
fummatrix of F is
S
22
I ·* Xx
1 + x
:9
, (Art. 10.)
with which proceeding as in the laſt Art. the required fummatrix will come out as
follows,
(0) X
29%
I
I
+ &c.
p+q. p + q. 2x+1
p + 2 q· p + 9.2x+2
n π
I
p + πq · p + q.2% +π
Where as many terms of this feries muſt be taken as there are units in, which will
then exhibit the fum of as many pairs of terms, or initial differences of the propoſed
feries, as there are units in z; the value of z being affumed at pleaſure.
เ
16. We ſhall now give a full exemplification of this method, by applying it to a
variety of feries; not confidering thoſe only of which the fummations have been given
by other writers, but extending it to thofe kinds of feries which have been hitherto
deemed incapable of any general expreffion for their fums. And this is effected either
purely algebraically, by the arch of a circle, or by logarithms, and that in a moft
elegant and fimple manner.
EXAMPLE I.
17. Let the feries to be fummed be
I
I
I
I
I
+
+
+
+ &c.
4.7
x x 3+x
I. 4 2.5 3.6
from Stirling's Treatife of the Summation of Series, p.-25.
By comparing the general term of this feries with that in Art. 13. we have p = 0,
P
C
m
9 = 1, m = 3, 1, and
n ~~I,
13
9
E 3
3, a whole number; confequently
the
20
THE SUMMATION
the fum of the feries may be expreffed algebraically. Subftituting therefore the above
values of p, q, m, n, and in the formula N, the fummatrix becomes
Z
Z
Z
+
+
;
3+3% 12+6x 27+92
and by writing the fame values in the formula L, we have the fum of the propofed»
I I
feries ad infinitum =
18
EXAMPLE II:
18. Let the feries propoſed to be fummed be.
I'
I
+
+
Τ
2.6 4.8 6. 10. + &c.
10
I
28.4+22
p. 117. From comparing the general terms,
か
​from De Moivre's Mifcellanea Analytica,
m
Hence as
75 2, is a whole
ท
q
we have p = 0., q = n = 2, m = 4.
n: = 2, m = 4.
number, the general fum of the feries may be had algebraically... Subftituting there- -
fore as before the values of p, q, &c. in the formulæ N and L, the fummatrix arifing
will be..
5x+3x²
32+48z+16 z²;
and the fum in infinitum =
3.
16.
EXAMPLE III
19. Let the fame feries be propoſed when the figns change alternately,
I
2.6
T
I
I
4.8
+.
6'. 10
I
8.12
+ &c.
Here, becauſe 2, an even whole number, the ſum of the feries may be obtained
algebraically (Art. 13.). For fubftituting the values defined by p, q, &c. in the for-
I
mula for the fum (M), there arifeth = the value of the feries continued in infini-
16
tum; and the fummatrix (O) becomes
2x²+3x
16.2%+1.2+I
in which as many pairs of terms, or initial differences of the propofed feries are. col-
lected, as there are units in z; which may be any whole number taken at pleaſure.
E. X A M P. L. E..
IV..
20. Required the fum of the infinite ſeries,
I
I
I
I
+
&c.
4.8
6.10
8.12
2+2% × 6+2%
Here
OF
21
CONVERGING
SERIE S.
་
Here we have p = 2, 9 = 2,. m = 6, n = 2, and thence = 2; therefore the
fum of the ſeries is given algebraically, being an even integral number. Subftitute
theſe values in the formule O, M, and the fummatrix arifing will be
x
Z
16+16%
I
but the fum in infinitum =
48
36+242
EXAMPLE
21. What is the fum of the infinite feries
6
6
+
2.7 7. 12
6
+
+ &c.
V.
6
−3+5% × 2+5%
12.17
Becauſe p =
3, 9 = 5, m = 2, n = 5, we ſhall have = 1; hence the alge-
braic fum of the feries is given. The values of p, q, &c. being fubftituted in the
formule N, L, the fummatrix of the feries comes out¨
-
3%
2+5%
;
and the ſum in infinitum = 3,
Which agrees with that of Vincent Riccati in his
Comment. de Ser. Sum. p. 20.
EXAMPLE VI
22. Let the feries to be fummed be
I
I
I
I
+
+
+
9.16
+. &c.
12:20
32.4+4%
3.8 6.12
Here we have p=0, q = 3, m = 4,
n = 4, and ≈ = 1, an integral number.
The general fum of the feries is therefore algebraical; the fummatrix (N) being
and the ſum (L) in infinitum = 1/12:
|:
2
12 +12.2.
EXAMPLE VII.
23. Let the feries to be fummed be
I
I
+
+
8.18 10 21 12.24 14.27
I
I
I
+
+ &c.
6+2x X 15+3%
Becauſe
22
THE SUM MATIÓN
Becaufe
(N) arifing is
6, 9 = 2, m = 15, n=3, it will be = 2; hence the fummatrix
I
X
+
3
8×8+22
and the fum (L)
I
A10
Х
I
8
100
+
I
10
11
I
10 × 10+2z
3
2. 5. 8
EXAMPLE
24. Required the fum of the feries
11
3
80
VIIL
I
I
+
+
5. 12
It will be p=0, 9 = 5,
10 15
m = 9,
T
15. 18
+ &c.
I
5%•9+3%
3, and thence. Subftitute thefe
values in the formule N, L, and the fummatrix' will be
5.9
+
I
2.2+2
Z
I
I
X
+
I+Z
3·3+z
I
and the fum =
9
I
5
Ι
I
I I
II
X +. +
小
​=
10
1.5
3.9. 10 270
EXAMPLE
25. Let the feries to be fummed be
I
3.6
I
I
+
6.8 9. 10
IX.
I
'I
+ &c.
12.12
7 32·4+22
Here it will be p = 0, 9 = 3; m = 4, n = 2; hence 2, a whole number,
3, π
and the fun of the feries is expreffed algebraically. Subftitute therefore theſe values
in the formulæ O, M, and the fummatrix arifing is
I
3≈ X
2
3·3+62
1
and the fum in infinitum
24
AL
I
6.6+6%
L
3
26. We might produce a great many more examples from feveral eminent Mathe-
maticians, as Goldbach, Taylor, the Bernoulles, and others; but theſe we have
already given, it is preſumed, are abundantly fufficient to evince the fuperiority of
our method to thofe who can judge of the fubject, in refpect of its elegancy, fimpli-
city, and generality. For it is obfervable that it is applied with the fame facility to
ſeries of which the figns change alternately from pofitive to negative, as to thofe
affected
1
OF
SERI E S.
23
CONVERGING
affected with conftant figns. And from the four laſt examples it is obvious, that it is
not neceffary for the factors in the denominators of the terms of the feries to be imme-
diately confequent to each other, neither need they be in the fame arithmetical pro-
greffion; but all the authors we know of that have confidered this fubject, have
thought one or other of theſe conditions abfolutely requifite for the fummation of the
ſeries to be exhibited in a pure algebraic expreffion. It muft however be acknow-
ledged that the fummations of all feries whatever of this ſpecies cannot be effected
purely algebraically neither by this procefs nor any other; but one moſt uſeful pro-
perty of our method is, that it is eafily determined by a proper fubftitution of the
values of the ſymbols p, q, &c. in the oecumenical formula, whether the propoſed
feries be fummible or not. For in feries of which the general term is
if π =
m
n
Р
9
I
p+qz.m+nz
come out a fraЯional number, or in feries with changeable figns,
and uneven whole number, the algebraic fum cannot poffibly be obtained, either by
this method, or any other exifting. And this is the indubitable mark, or true cri-
terion of the poffibility of the algebraic ſummation of a feries; a property which will
be found to be of the greateſt utility in abſtracted mathematics. The fummations of
thofe ſeries of the above form which do not admit the algebraic fum, may neverthelefs
be obtained from the fame cecumenical formula, by the affiftance of logarithims, or
the arch of a circle, which will be manifeft from a few examples; to theſe we ſhall
therefore now procced.
EXAMPLE X.
27. Required the fum of the infinite ferics.
I
•
I
I
+
+
+ &c.
I 2
4.5
7.8
Becauſe p
I
-2+3x-1+3%
I
2, 9 n = 3, 11 = 1, it will be = a fractional number,
介
​3
therefore the fum cannot be expreffed algebraically. But by fubftituting thefe values
of p, q, &c. in the cecumenical formula, it becomes
*
I
; in which let y be written for x
and
3
2
3
3
x 3
·
I-X
* 3 . I-X
the fum of the feries will be
S
I + y + y²
But this is equal to 4 the arch of a circle of which the rad. is 3, tang. y + { ;
3
F
which
24
THE SUMMATION
✓ and tang. Зу
which by a proper correction becomes arch, rad. 3,
4
=
3
4+2y
M.king therfore y 1, agreeable to the method before obf rved, the required fum
of the feries will be truly expreffed by the arch of 40° of the circle the rad. of which
is & √3.
2
EXAMPLE
28. Let the fum of this feries be required,
XI.
3.3 5.9
I
I
I
+
The general term of the feries is
I
I
+
&c.
7.27
9.81
I
I+ 22.3≈—I
x p x
& South (Art. 7.)
9
which belongeth to the formula
In which the values of p, q, and B (drawn from the general term) being fubftituted,
there arifes
1
3
2
S
3+x. √x
for the fum of the given feries.
Puty, and the expreffion for the fum becomes
S
3 j
3+ y²°
But the fluent of
this expreffion is the arch of a circle of which the rad. is 3 and tang. y; which evi-
dently needs no correction. Maki g therefore y = 1, as before, the fum of this
feries continued ad infinitum, will be exprcffed by the length of the arch of 30° of
that circle of which the rad. is √ 3.
EXAMPLE XII.
29. Let the feries propofed be
I
I
I
I
+
I. 2
+
+ &c.
3.4 5.6
−1+2≈.2Z
Here we have p =
1, q = 2, m = 2, n = 2, and π = ½.
9
fubftituted in the cecumenical formula, and reduced, there ariſeth
Thefe values being
I
2
S
I+ vx. √ x
In which for x write y, (and let L. always denote the hyp. log. of the quantity
connected with it,) and it produces
= L. ry = (by making y = I as
EXAM-
before) L. 2, the fum of the propofed feries continued ad infinitum.
OF CONVERGING SERI E S.
25
EXAMPLE XII.
30. Required the fum of the feries
Becauſe p I, 9
I
I
I
+
+
3.5 5.8 7 7. II
I
+ &c.
1+2x. 2+3 =
I
Subftituting there-
= 2, m = 2, n = 3, it will be ≈ =
fore theſe values in the cecumenical formula, we have
/
x23x
S√x - * - S = = = =
I
I
In the former part of the fluential put √x=u, and it is changed into 2
of
3
24.
But 2
S
i
= hyp. log. of
1 — 11 ²
I tu
1
S
14
I น
; therefore, by restoring the value
И, the fluent of the former part of the above expreflion becomes L. 1 Vx
✔x, which needs no correction. Again, in the latter part of the fluential make
√xy, and it becomes
ولا
L. [+y+y
I
-L.1-y+ L. i+y+y 2 Q, which alio needs no correction.
2
Hence the above fluential expreffion becomes
L.i+vx L. 1 − √x + L.
I
I
and it becomes 3 3 Siy
3
y²; but
2
3 SIL
I
I
= L.
+
I
y
2
I
2 Q, (Q being the arch of a circle, rad.
✓ 3, and tang.
2
3 y
4+2y
3
3
L. 1+ √*+ √x² + ? Q+
༡
33
2
I
I
- — L. 3
2
2
2
I
tang. ) + L.
*²+2Q+. And in making = 1, we have L. 2
I
½ + 2 Q = 2 V (V being the arch of a circle, rad.—3, and
?
T
=arch of 60°^ + L. 一
​√3
2
3
2
EXAMPLE
31. Let the feries to be fummed be
XIV.
I
I
I
I
1
1.4
2.5
+
3.6
+ &c.
4.7
•
3+=
F 2
Here
26
SUMMATION
THE
Here will be an uneven whole number, and therefore the fum of the feries cannot
be exhibited algebraically (Art. 13, 26.). But by fubftituting the values of p, q,
m,, in the œcumenical formula, there arifeth
x³ x
;
I
I
3
3
S
which is L. 1+*·
2,2
ર
*
+
9
3
And making I, we have
x
2
L. - 2
5
3
18'
the fum of the feries continued ad infinitum.
NOTES,
OF CONVERGING SERIES.
27
NO T
T E S,
EXPLANATORY,
ON SECTION II.
ART.
9.
The equation
p :
F (= = √x194)
q
being multiplied by xm:n-pq-1x, becomes
I
卫
​P
+i+
I
x 9
n
11
Ptq
x n x
p+q
+ I
x
m + n
pte
+I
p+q
H+
H
xp:9+2
+29
+ &c.
P+9
P
??
+2+
x 9
7
9
#1
1
I
*.
p+29
+ &c. And the fluent is
+2
+ &c.
m + 2 n
p+29.
n
+ &c.
n
nxm:nti
p+q. 1+1
+1
nxm: " + 2
p+29.m+288
xm:n+1
Hence S
1
Fxm:mp: 4-s
n
F + q
• 771 -- 72
+
+ &c.
xm: » +2
Þ+29. 211+21
+ &c.
And in like manner may the other expreffions H, I, &c. be found.
ART.
28
SUMMATION
THE SUM
ART.
13.
min-piqu] Ÿ
x p : q x
IF X
To reduce the expreffion Sx
let the former part of
the fluential be put, and the latter, then it will become
S&S, which
ù,
by Art. 11. is = v
S
й
Svù
ข
vù; hence arifes
q n
Si
I
u
q n
reſtoring the values of v and ú,
But
x
ทา
n
fince
#
771
ni
Р
I
9
x 9 x
I
x
n
9
n q
in
р
IF X
n g
772
P
n
22
9
9
x 4 x
IF X
7
+
IF X
р
9 x
22
9
x 9 x
I
x
12
X
is =
P
I F t
9
ㄧ​ˊ
X
P
9
I
mn
Р
is conftant, the latter fluential becomes.
n
vi; or, by
;
therefore
9
x m : n− p q
9
x p : q x
I
m
р
IF X
712 P
xm: xx
IF X
Now
nq
ng
22
12
9
put x = 1, and the laft expreffion becomes
I
xP: 9
m
P
xm: n x x
! FX
ng.
n
9
ز
in which it is obvious that when the fluen-
tial is expanded, if the negative fign be uſed in the Denominator, the refulting terms
will be affirmative; but if the affirmative fign be retained, the terms will be alter-
nately + and
And in this laft expreffion
Hence
xP: 9
xm:n Xx is xp 9 x
:
xm: nx
ni
P
xmin
= I
X x P : 9 x = 1
*
X
q X X
X x P 1 9 x
92
x P: 9
I
XT X XP: qx.
I
пят
S
I xn. x P : q x
I-X
nq π
OF CONVERGING SERI E S.
29
I
T
n q π
S
X xp: 9 ; but the fractional part
I
χ
I
X
I-X
may be thrown into a feries thus,
1-x) I-X™
I *
x-xx
x-x³
XT
(1 + x + x² + *³ + &c.
continued till the exponent
of x becomes ==
I,
where it will terminate
in terms.
X
Xπ
x³-x4
x+- &c.
From whence there arifes
I
T
nq π
I
пять
And the fluent is
=
-
I
π q
I
મ
元
​n q π
I
1 + x + x² + &c. × xp: qx; or
S1
S
x P : q x + x P : 9+1 x + xp:q+2 x + &c. xp:q+- *.
xp: 9+1
:
+
p+9:9
qxpiq+1
+
p + q
x4:9+2
P+29:9
9xp:9+2
P+29
xP:q+=
+ &c.
p+=9:4
+ &c.
xp:q+1
+ &c.
Nπ
P+9
ART. 14.
The Expreffion
S
xm: n―p: 9—1 x
s
I
1 − i z. x P : 9 x
q
1 X
being refolved into two diftin&t fluentials by Art. 11. becomes
772
P
9
X 7
m
17
11
P
9
S
I
•
q
Þ
12
x 9 x
S
X
9
I
x 7
X
m
P
I
3
P
x 9 x
1-X
S
X T
A
n
9
X
ala
-x~. x 9 x
•
I - X
11
1
મ
30
THE SUMMATION
P
I
X π
1 − x~
x 9 x
T
I-X
S
x =
I
I-A
(by making I without the fign)
р
P
I
75
I
S
nq π
J
זין.
I
I ༢༩ ཝ
X 4 X
I
I t. I
x P: 1 x
7
.४
And this expreffion may be farther reduced as follows,
= (multiplying by 4)
n
I
First,
÷ S
1 x. I
=-
xx
xp : q
x
•
but
I
Xx
I X
I - V
1 + x + x²+ ~·³ +x++ &c. x-I,
which multiplied by xp becomes.
I xπ
1- X
9 ✰
S =
I - X*
X x P : q x. I−x~;
I-X
× xp: 9 x = x P : 9 x + x p: q+¹ x + xp: q + 2 x + &c. xp: 7+7~1.
This again multiplied by 1-≈ produces
I
xp: q x. I
xx = xp: 1 x
I
xp:q+x ÷ + xp: ? + ¹ x
1
xp:qt≈ti ¿ +
&c. xp: 9+ X
And the fluent will be
xp: q + = + x−1 x.
p: q+1
P: 9+1
or,
p + q
xp: q+x+I
p: q+z+I
9xp: 9+x+1
p + q + z
+ &c.
xp: q+#
P: 9+π
7″
qx
+ &c.
p + qo
xP:q+~+x
p9+π+2
9xp:s+x+≈
p: ?+*+z
which multiplied by
1
there arifes
>
123 76
I
X
N T
xpiati
x1:9+x+1
9
+ &c.
P+ q
p+9.2+I
p+qx
wp: q+x+x
p+q.π+z
and making I, we get
a * =
I
I
X :
ทช p + q
I
I
+ &c.
p+q.x+1
P + q π
I
p+q.π+Z
or, by reducing each fucceffive refidual to a common denominator
I p+q.2+1
X:
p + q
•
+·lic.
p+q. π + xp+qπ
NT
p+q.p+q.%+1
•
11
I
72 π
OF CONVERGING SERIES.
I
q z
X:
+ &c.
N T
p + q · p + q.2+1
א
I
X:
+
&c.
NT
p+q. p + q. 2+1
T
92
p + qπ · p + q = + z
I
T
p + q = · p + q •π + ≈
By the fame method is the fummatrix O (Art. 15.) inveftigated.
The terms to be compared are
I
ART. 17.
I
2.3+%
p+qz.m+nz
From whence appear the values of p, q, &c. as in the example.
Theſe ſubſtituted in the fummatrix N, Art. 14. there arifes
རུ་
I X Z
1
I
X:
+
I
+
I X 3
0+1.0+1.≈+1 0+2.0+1.2+2 0+3·0+1:%+3
I
Ι
1
X
+
2.≈+2
11
L
3
fummatrix.
=
The fame values being put in the formula L, Art. 13.
+
3·2+3
Z
R
Z
+
+
the
3+3%
12+0≈
27+9%
I
1
I
I
I
give
3
X
+
0+ I 0+2 0+3
+
X 1 +
3
2
the fum.
m
Hence it appears, that as
n
q
+
I
I
I I
I I
X
3
3
18
(3) is a whole number, the fum of the
propofed feries is obtained purely algebraically by taking (3) terms of the formula
L. And in like manner is the fummatrix of the given feries found by taking ≈ (3)
terms of the formula N. Nothing therefore can be more obvious than this method
of fummation. For when any feries of this fpecies is propoſed to be fummed, we firſt
find the general term thereof, which is readily done by a bare inſpection; then by
comparing this general term with that of the general feries, we derive the values of
m, n, &c. and from thence appears the value of, which if it comes out a whole
number, indicates that the feries is fummible in finite terms; if a fraction, it denotes
that the fum of the feries can be had only by logarithms or circular arcs. In the fum-
matrix it muſt be obſerved, that the quantity z is no ways affected by the value of *,
for z is an indefinite number, and in feries of which the figns are poſitive, whatever
whole number be affumed for it, the formula reduced gives the fum of that number of
G
initial
3.2
THE SUMMATION
initial terms of the feries.
given feries was required.
For inftance, fuppofe the fum of the first 8 terms of the
Here x = 8, and the fummatrix becomes
I
758
X
+ +
-
8
3
9
20 33 1485
And with the fame facility may any number of intermediate fucceffive terms be had,
by the difference of their fummatrices.
Suppoſe the fum of 21 terms of the above feries was required, commencing at the
16th term.
The firſt value of x is 15, and the fecond value 15 +21= 36, hence.
Z
12
I 2
+
37 76.
+
ulti
+
2 3/0
12
4055
+
117
7344
5
4809
34
54.
82251
298210509 is the fum required.
604051344
-
ART.
19.
16
And the difference
as.
Here it must be obferved that in fubftituting the values of p, q, &c. for the fum-
matrix, only fo many terms of the formula muſt be taken as there are units in
before; but in reducing the numerical fummatrix, whatever number be affumed for
z, it will give the fum of twice that number of initial terms in the given feries. This
is obvious; for here denotes the number of initial differences, which must therefore
be twice the number of fingle terms.
In this example the terms compared are
+
22.4+22
H
I
p+qz Xm+nz
;
from whence we have the values of p, q, &c. which being ſubſtituted in the formula
M (Art. 13.) there arifeth
I
I
I
I
I
I
X
X
16, the fum.
2 2
2
4
4
4
And the fame values fubftituted in the formula O (Art. 15.) it becomes
J
1
Z X
8x+4
162+16
the fummatrix.
8x²+12%
2x²+3%
8x+4 X 16x+16 16 . 2+1 · 22+1
Now affume for z any whole number at pleaſure, ſuppoſe 2, and the laſt expreffion ·
becomes, which is the fum of the first four terms, or the 2 (z) initial differences
120
of the given feries.
1
3
ART.
OF CONVERGING SERIES.
33.
ART. 20%
The terms compared are
I
2+2x X6 + 2 z
1
I
p + qz X m +nz
☛
from whence the values of p, q, &c. are known, and thence 2, which ſubſt:-
tuted in the general fummatrix for feries with changeatle ſigns (O) there ariſeth
2
•
22
X:
2. 2
I
2+2.2+2.2z+I
I
2+2.2.2+2.2≈+2.
27
4.4%+4
Z
6
4≈ +6³
for the required fummatrix.
The fame values being written in the formula M, give
I
I
X
I
I
I
= X
4
I
12
489
the value of the propofed feries ad
2 2 2+2 2+2.2
infinitum.-
To exemplify the fummatrix.--
Suppofe the fum of 100 terms of this feries
was required.— -Here z 50, the number of initial differences, hence
50
50
4.X 204
6 X 206
875
42024
is the required fum,
ART. 21.
Here the general terms compared are
I
- 3+5% X 2 +5%
I
p+qz Xm+nz
The values of p, q, &c. being fubftituted in the formula N (Art. 14.) there is
produced
I
5
5
3x X
-3+5 X-3+5.2+I
ra
=
2
•
5x+2
10≈++
which multiplied by 6 the common numerator in the feries, it becomes
6≈
"
༢༤ the required fummatrix.
4+102 2 +5≈
The fame values being written in the formula L, produce
I
I
X
—which multiplied by 6, gives, the whole value
5 -5+3 ΙΟ
of the propoſed infinite feries.
G 3
34
THE SUMMA TI O N
If the preceding examples are attentively confidered, it will be wholly fuperfluous
to attempt any illuftration to the four following ones; we fhall therefore proceed to
T
ART. 27.
From the values of p, q, &c. found as before by comparing the general terms, we
have a fraction; which indicates that the fummation of this feries cannot be
effected purely algebraically. By writing the values of p, q, &c. in the cecumenical
formula, it becomes
which is =
I
|
3
S
४
3
im
x x
I
3.3. //
x
I-X
Ι
2:3
x
X 1
3
S
X
X
1: 3 X 1 - X
I
3
2:3
43
3
= x
Xx
× ×³ = y², and x =
-S
3 y² j
s
•
تو
تو لو
I
у хў
=
I — y³
3
we have y³x, and..
And if y be put = x
3y²; hence it becomes
3
S
but
I
3 y² j
y². 1-y
3:
becomes
3
Si
I
3
· 1 + y + y²
I + y + y²
(which appears by actually dividing), the fluential therefore
Now the fluent of this expreffion is eaſily inveftigated; -for
multiplying the fluxion by 4, it becomes
34
3+&+y+y² √ √ 3 +
✓ . v . j
+·y + y²
&
플​~3
√3. √3.j
=
; the fluent
22
1 √3 • // √ 3 + 4 + y + y²
of which (by Emerfon's Flux. p. 239.) is the arch of a circle, radius ½ ✓ 3, and
+y+y², or y+; confequently dividing this arch by 3, or multiplying
tangent
3
it by, gives the fluent of the expreffion
But here the fluent evidently
wants a correction, for when yo, it becomes arch of a circle, rad. √3, and
½
tang.; therefore arch, rad. 3, tang. y + - arch of the fame circle,
√ 1/1/2
tang., is the correct fluent. This however may be expreffed much neater; for the
3y
tangent of the differences of the above arches is
4+2y
1 + y + y²²°
उ
which may be deduced from
Emerfon's Tiig. p. 24. prop. 9. where we have this analogy
as rad.² + y + 1 × : rad.² :: y + - the tangent of the differences
1 1/
of the arches, that is
as 1+ y
y:
22 y
1+ y
3y
4+2y
; hence when y, the above
fluent becomes arch of a circle, rad. 3, tang. . Now the arch correfponding
to
OF CONVERGING
3.5
SERIES.
to this tangent and radius is 30 degrees; for it will be, as√3: 1 :: rad. : tang.
that is (rad. 1) { ÷ 3 =
= {√3
of 30°
I
√3
='57735 the natural tangent of 30°; but }
40°, hence the fluent is truly expreffed by the length of the arch of 40°
3; which is therefore the fum.of the propoſed
of that circle of which the radius is
infinite feries.
ART. 28:
By Art. 7. we have (writing & for m)
I
9
B+ x
or, by multiplying by ß
В
S
9
β
or,
xp: 9
93
x piq x
xp:9+1
=
p + q. B
xp:9+2
P + 29. B²
xp: 9+z
+ &c.
P + qz. Bx
Xp:9+1
1 +9
xp÷9+2
P + 29. B
xp:9+x
+ &c.
I
=
p + q
+ &c.
I
P+29. B
p+qz.8z
B+x
xp: q x
S* B+X
Hence the general terms compared are
I
− 1 + 22 · 3²~}
I
P+qz. B²-1
p+qz. 8² = 1;
2, and 6 = 3, which written in the above fluential, there
from whence p = 1, 9 =
arifes
=
3
x - - x
3
-
2
3 + *
2
3+* . √ ≈
Now the fluent of
39
is known by inſpection only; for
3+y
rri
is the fluxion of
rr tit
the arch of a circle, expreſſed in terms of the radius and targent; hence by compari-
fon, the fluent of
(=
3 j
3+ y²
√3.√3.3
is the arch of a circle, rad. 3, and
√3⋅ √·3+y²
tang. y. And if yo, the expreffion for the fluent vanishes, the arch and its tan-
gent being cotemporaneous; and therefore it needs no correction. And when y is
expounded by 1, the arch of the circle corresponding to rad. ✔3 and tang. 1, may
be found thus,
I
√3
='57735 the nat. tang. of 30 degrees.
as √3: 1: I :
Confequently the fun of the given infinite fèries is truly expreffed by—the length
of the arch of 30 of a circle when the radius is 3.
It is obfervable that the fummation of this feries was firſt propoſed by Dr. Halley,
with a view to effect the quadrature of the circle; he having difcovered that the fum
of.
35
SUMMATƯỞN
THE
of this feries multiplied by 2 3 would give the perfect area of the circle when the
radius is 1. Great were the expectations of the Literati at the time the Doctor pub-
licly propoſed is of accompliſhing this very important point in the mathematics. Se-
veral eminent men attempted it; but without fuccefs. Among thefe was Stirling,
who, though the most affiduous to effect it, was obliged to reft contented with an
approximation only to the fum. We muſt indeed acknowledge that the fummation of
the ſeries here given does not much conduce to the completing of the quadrature
of the circle; but it plainly exhibits the foundation or reafon of the Doctor's method,
which I believe was not for a long time known to any but himſelf. For fince the
length of the arch of 30° is th the femicircumference, it is obvious that if it be mul-
tiplied into the diameter, the product will be the area of the circle, and therefore
the length of the arch of 30° (which is the fum of the propoſed ſeries, rad. 1.) being
multiplied by the diameter 23 will be the area of the circle. But the area of a
circle, rad. ✔3, or diameter 2 V3, is the whole area of a circle when the radius is
I ; for 2√3X78539 &c. = 9'4246 is the area of the above circle, one third of
which is 31415 &c. the area of a circle when the femidiameter is 1.
J
•
ART. 29.
The terms to be compared are
I
- 1+2 ≈ X 2 %
I
p+qz Xm+nz
from whence the values of p, q, &c. appear as in the example, and
ol
I
I
+
a fraction.
2
2
2
m P
n
L
Theſe being fubftituted in the cecumenical formula, it
becomes
I
2.2. //
I
2
S
I-
X
√x
I
xo Xx
I
S
I
1 - X
I
vx.x
I
√x.x
I-
ހ
*; but
2
I−X. √x
*^ • *— I
I
X
x²+x
I
.. the laſt expreffion becomes
= √ √ 1 + x
2
j
And the fluent of
=
2
S
is well known to be the hyp. log. of i+y, or by writing 1
Ity
for y, the hyp. log. of 2.
1
This
OF
37
CONVERGING SERIES.
This feries was invented by Lord Brounker for the quadrature of the Hyperbola,
and publiſhed in the Philofophical Tranſactions, Number 134; being the firſt piece
ever written on that ſubject.
ART.
30.
The values of p, q, &c. fubftituted in the cecumenical formula, produce
I
3.2
S
x
플
​I-
I
2
//.
S
X 3
=3
S
I- X
3.2. 1-*
S***.
I
And if u be put for √x, x is = u², and ¿ = 2 u u; hence the former part of the
fluential becomes
S
2 u² ú
I
·
which may be evidently refolved into
(becauſe
2
`2 ù−2 ù +2 u² ú
Szi-
2 ú-2 u² ú
1 — u²
2
й.
29
2 by divifion)
11
I
2ů
·u²
2 Ú-2u² ù
й
2
I
น
ƒ zü
2 ů = 2
28.
2
I
1-2
i
-1-
And the fluent of
is had by form 6th, Emerfon's Fluxions, where by compar-
ing the fluxions we have a 1, B = 1, " = 2, and z = z; theſe values written in
the correſponding fluent produce
L
I+u
X vulg, log..of
>
which multiplied by 2 becomes L × vulg. log. of
2
1-2
I + u
1+u
I + vx
= hyp. log. of
therefore, by reſtoring the value of u, L.
K
I
I
√x.
2 x is the fluent of the former part of the expreffion.
3
And when y is written for x in the latter member of the fluential, x will be.
y³, and 3y²; hence it becomes.
#
which may be transformed into
3 y* j
£325,
ƒ 399 - 31 j + 31 j = S 3ÿÿ
I — y³
3yy
1 — y³
319-3y
1-y³
S322 - £35=3f 2-1,.
Now
38
THE SUMMATION
Now the fluent ofÿ
may be had by form 8th, Emerfon's Fluxions, as follows,
Compare the Auxions, and there will be a = 1, B = −1, = 3; and in order
n
I may be 1, the exponent of y in the propofed fluxion, make - " = 2,
that
หฺ -
2
then is
or ♪ = 2, λ = 3;
እ
3
a = fine of 120°
=
A =
Hence
1, B =
I
I
180
እ
hence = 60°, and z = y = x.
y;
Alfo
(rad. 1.) √, s = cofine of 120' = -1; and by cafe 2d.
25 = +1, C = 2⋅ a = 2√2 = √3, D&c. = 0.
λ
3 · 1
.3-2
= the common multiplier
I
उ
into every term of the fluent; but as the given fluxion is multiplied by 3, it becomes
+1, which therefore does not affect any of the terms. Confequently the fluent
will be
-
LX log. of 1-y + L X log. of √1+*+*˜¯
of the arch of a circle the fine of which is
V3 X NX into deg
x
2/2 1/3
√i+*+*
It may not be amifs
here to obſerve a few things to the young reader previous to reducing this expreffion
to that given in the example, which will not only render this much clearer, but
may be of fervice in educing other forms to fimpler terins.The hyperbolic
logarithm of 10, denoted by L, being multiplied into the common logarithms pro-
duces the hyperbolic logarithms.-The log. of a quantity under a radical fign is the
fame as the exponent of the radix multiplied into the log. of the root.The num-
ber denoted by N multiplied into the degrees of the given arch of a circle gives the
length of the arch when the radius is; and confequently if the length of this arch
be again drawn into fome number, as 3, the product will give the length of an arch
✔3,
containing the fame degrees to rad. 3; all which will be plain from a little confi-
deration. Therefore the above fluent may be expreſſed thus,
hyp.log. i−y + ½ hyp. log. 1+y+y² √3 × into the length of the
•
arch, rad. 1, and fine
or the arch, rad.
3, and fine
Vi+y+y
√i + y + y²
3
2 y
Vi+y+y
3y
or twice the arch, rad. 3, and tang.
Z
; which laſt term
>
4+2y
is inveſtigated thus,
3
As cofine is to fine fo is radius to the tangent; but to rad. ✔3 and fine
y
i+y+ y²
the
OF CONVERGING
30
SERIE S.
the cofine is
is V
3
2 y
I+y+ y²
2 y²
y
hence
3
√3
3:
Vi+y+y
(by multiplying, reducing to a common denominator and dividing)
?
1 + y + y²
√3.3
3 + 3 y + 3 y'
3+3+3y-2y'
3
y
vity + sy
3 y
4+2
the tangent to rad. 3; confe
I
2
quently twice the correfponding arch must be equal the length of the above arch when
the rad. is 3.
But it may be obſerved, that the fame conclufions as in the example may be
brought out immediately from the fluent refulting from the tables, without finding an
expreffion for the tangent to a different radius. For making y 1, according to
the prefcript, the laft member of the fluent becomes 3 x into the arch, rad. 1,
and fine, that is -the arch of 30°, rad. 3, which must evidently be the length of
the arch of to in a circle of half that radius, that is √3.
✓
I have been the more particular in the method of finding this fluent, as fluxionary
expreffions of this kind depending on the meaſures of ratios and angles are fomewhat
intricate; and the applications of Emerfon's forms (7th and 8th) are often mif-
taken, both in the fubftitution and the figns.
ART. 31.
By writing the values of p, q, &c. in the cecumenical formula, we get
I
༨
I X I X
1
I
20
X x
=
I + *
3
S
I
1 + *
3
S
I + *
The fluent of the former part
is expreffed by the hyp. log, of. And the
may be refolved into
latter part
I + t
X
+1
I+
I
x² + 1
=
*
I + x
*; but
1+3
=起
​I + x
by actually dividing, therefore the fluxion becomes
1 + x
the fluent of which is
2-3
3
اني
+*; confequently the whole fluent will be
I
I
hyp. log. 1+*
3
3
38 / 03
1
x
+ it'
2
H
이
​1
+ 1, as appears
++
hyp. log, of
hyp. log. of 1+* =
3
001-
40
THE SUMMATION.
1
1
3
I
+3
x
L. 1+*
+
+ L. 1+x =
3
3
3
L. 1+ *
+
1
3
9
6
3
The fluent may alſo be had by Einerfon's
forms 6th and 7th, without any transformation in the latter part of the
fluential.
The ſummation of this feries is faid to be impoffible by Stirling, and ſeveral
other authors.
1
SECTION III.
OF CONVERGING SERIES.
41
SECTION III.
PROP. VIII.
32. To find the Sum of a Series, when the Denominators of the Terms confift of three
fimple Factors, as
I
p+q.m+n.rts
I
+1
p+39. m+3n.r+35
I
p+qz. m+nx.r+sz
I
+
p+29. m+2n.r+25
± &c. the general Term being
The formula for the fum of this feries (Art. 9.) is
H =
F
=
I
S
Gris-mix-. But G =
SG *
S
x p : 9 x
22
SF
Fx*:
—9—1x, and
✰,
; hence by fubftitution we have
I
H =
X
9-1
xp:
5
77L
9 x
* F *
s—m: 2-I
x
S
ng x
m
n
nq X
n
9
Let the former part of this fluential be denoted by P, and the latter by Q; each of
which being refolved into two parts (Art. 12.) and making
(P)
(e)
*
m
= 5,
=
S
9
12
x
x P : q
M2 X m s bu
-2
W
ms
1-2 X 2 s b u
وله
m
P
and therefore
1119 ; we get
11
9
S
IFN
S
IFN
xm: nx
IFX
ngs π X π-
I
ngsw X ---
H 2
And
42
THE SUMMATION
And by making x = 1 without the fluential fign, they become
(Ý)
I
NG ST X π--
I
S
I X
π
X xp : 9 x
(é)
nqs w x π
- W
S
IFX
Wherefore P-Q will be the general fum of the propoſed ſeries in infinitum.
33. Hence when the figns of the propofed feries are pofitive, the fummation may be
exhibited algebraically as often as and w are any whole numbers whatever: but if the
figns change alternately from pofitive to negative, the algebraic fum can be obtained.
only when
are even whole numbers. For from the formula
π
(P) arifeth
and
I
Nqs π X T
the fluent of which is
xp: q+π Π
B+πα
f
4
1+x+x²+x³ +x²+ &c. **-1 × xp:9 x ;
I
x:
N₁S π X T-W.
xpiq+1
p + q
+
xp:9+2
2+29
+ &c.
; and making x1 it becomes.
1
(P)
I
I
I'
+
+
NS π X π — W
p + q p+29
p+39
And after the ſame manner may the formula
+ &c.
te changed into
I
(2)
I
X :
95 w X
I
m+ n
I
I
+
+
+ &c.
ω
m + 2 n
m² + 3 · n
I
M w n
therefore P-Q is the algebraic fum of the feries, the general term of which is
Ι
p + qz. m + nz. r+sz?
where as many terms are taken as there are units in and w reſpectively.
34. But when the figns of the propofed feries are alternately changeable, and 7,
are even integral numbers, the formula P becomes
Q
I-
NQ ST X T
I x + x².
x³+xª—x²+&c. x*−1 × xp:9x,
from whence, by taking the fluent, and making x = 1, there arifeth
(P)
1
NST X π-w
I
X:
p+q
I
I
+
p+2.9 P+39
After the fame method is the formula Q changed into
i
+ &c.
P+49
P+9
t
I
I
I
I
(Q)
I
X:
+
+ &c.
q s W X π
m + n
m + 2 n
ω
m+ 3n m + 4 n
I
m + w n
Wherefore ❞
is the algebraic fum of the feries; as many pairs of
terms being taken as there are units in and a refpectively.
35. The
OF CONVERGING
43
SERIES.
9
I
will be
I
H =
S
35. The fummatrices of theſe feries may alfo be derived as follows.In the
formula G (Art. 9.) ſubſtitute
(Art. 9.) ſubſtitute for A the value of the firft fummatrix
I
- xx X x P : 9 x
for feries affected with pofitive figns (Art. 10.), and it
I
S
xp
1 - xx X x P : q x
X
:9-1 x
ngs X π
9
ω
T
ngs X T
xr: s―m:n-1 x
Now, let each member of this expreffion be reſolved into two fluentials, making x = 1
in the finite quantities arifing, and we have
S
I
Xx min
x
1-X
H =
I
Nqs T X = - W
I
ζω Χπ
W X π-W
ngs w
and thefe by a farther reduction produce
(M)
- (N)
I
Nqs π X T
−2 X 1-xXxp: x
X I
;
S
- xx X x P : q x
I
I - X
S
I
x x x xm : n x
I
+
I
X
ngsw X 5 -
fi
S
I - X
༤ *
X XT5 X
1-X
- ш
n.qsw X π- W
S
S
Ι *
1.- xx X 1
1− ) « X xm : nx
; which again by divifion and
I-X
ヨード ​XE
3 - 26 X
Sx F
S
x p : 9 x + x p:? +1 x + &c. xp:7+−1 j
x pista x
xm: n x + x":n+1 x + &c. xm : n+x−1 †
X
I
multiplication become
(M)
I
N Q S T X π
xp:q+z+1 x &c.
(Ń)
I
ngs w x π
xm:n+z+1x &c. The fluents of theſe terms being taken in order, and in every
term x expounded by 1, there arifeth
(M) :
9 Z
#
(N)
NST X K-W
1 %
M-L x ms b
ах п
I
I
X:
+
+ &c.
p + q · p + q⋅ Z+1
.
p + 2 q. p + 9.2+z
I
p + #q. p + q. Z+™
x:
X :=
I'
I
+
=+ &c.
m+2n.m+n.~+2
m+n.m+n.≈+I
I
m+wn.m+1.≈ + @
Therefore
44
THE SUMMATION
Therefore the fummatrix of thefe feries will be M-N; where as many terms muft
be taken as there are units in # and w refpectively.
36. After the fame manner may the fummatrix be obtained when the figns of the
ſeries are alternately changeable from pofitive to negative, provided ≈ and w be even
whole numbers. For fince the general fummatrix of thefe fpecies of feries may be
expreffed by
A =
9
S
I - x²≈ X x p : 9 x
I+x
(Art. 10.j
if in the preceding formulæ M, N, we write every where 2% for %, and take the
terms alternately poſitive and negative, we ſhall have
(M)
292
3-1 x usb
X:
I
I
p + q · p + q.2% +1 p + 29. p +9.2x+2
I
p + q · p + q2z+π
+ &c.
(N)
2nZ
X:
qs
W X π-W
I
m+n.m+n.2z+ı
I
I
+ &c.
m+2n.m+n.2x+2
m+wn.m+n.2z+w
hence M-N will exprefs the required fummatrix, in which as many pairs of terms
of the given feries will be fummed as the index z contains units; as many terms of
M and Ń being taken as there are units in and reſpectively.
T
EXAMPLE I.
37. Required the fum of the feries
I
I
+
I
• 2
•
4 2.3.5 3.4.6
+
+ &c.
T
I
2 X 1+x × 3+z
It will be p = 0,9 = 1, m = n = s = 1, r = 3, and ✯ = 3, w = 2.
2. Wherefore
the algebraic ſum of the propofed feries may be obtained, by ſubſtituting the above
values in the formula P-Q, and M-Ń, refpectively. From the former arifes
?
7
2
18'
the value of the whole infinite feries; and the latter gives 7 × :
2
6
}+%
I
I
the fummatrix for terms.
2.2+z
3·3+≈
EXAM-
OF CONVERGING SERIES.
45
EXAMPLE II.
38. Let the feries propofed be
I
I
I
I
+
+
+ &c.
2 + 32 • 1 + 38.4+*
= =,
1.4.7 4.7.10 7.10.13
Here it will be p = −2, q = n = 5 = 3, m = 1, r = 4; and therefore
w = 1. Theſe values being written in the formula P–Q, and M-Ń, give refpec-
1
24
tively, the ſum ad infinitum, and Z- X
6
I
I
the fummatrix.
1+3%
4.4+3%
1
+
EXAMPLE III.
39. Let the feries to be fummed be
I
I.2.3
+
2.3.4 3.4 · 5
I
I
+ &c.
2 X 1+2 × 2+z
Here p = 0, m =
n = q
n = q = s = 1, r = 2, and thence π = 2, w = 1.
1. Where-
fore by placing theſe values in the formula ❞ – Ű, and Ń - Ń, there arifes
I
༤ .
Z
for the fum, and
for the fummatrix..
4
2 X 1+Z
2² × 2+2
EXAMPLE IV.
40. Required the fum of the feries
I
2.6.10
of which the general term is
4.8.12 6.10.14
I
I
+
&c.
1
2 ≈ X 4+22 X 8+2≈
= 4;
=
Hence p = 0, q = n = :s = 2, m = 4, r = 8, and therefore =
which, being even whole numbers, indicate that the propofed feries may be fummed
algebraically. Hence by a proper fubftitution of the above values in the formula
P–Q, and M—Ñ, the fum ariling will be
22 |
X
32
I
5
and the fummatrix
2
3
I
I
I
+
+
1+22
2.2+2x
3.3+20
4.4+
LX Ali-
45
THE SUMMATION
EXAMPLE
41. Let the feries propoſed be
I
I.9.10
I
+
I
2. 12. 12 3. 15. 14
+ &c.
V.
I
≈ X6 + 3x × 8+2%
Becauſe pó, q = 1, m = 6, n = 3, r = 8; s = 2, we have ≈ = 4, and @ = &,
both even whole numbers, and therefore the propofed feries admits of an algebraic
expreffion for the fum. Which appears as before by fubftituting the above values in
M
the formula -, from whence arifes
5
.8²
the fum in infinitum. The fame
values being written in the formula M-N produce
Z
I
X
24
1+22
I
I
2 X 2+2 Z
3 X 3+2%
the fummatrix for 2 z terms.
+
1.
4 X 4+22
EXAMPLE vi.
42. Let the feries to be fummed be
I
I
I
2.0.8 +3.8.10 + 4. 10. 12
+ &c.
I
1+z.4+2%.6+2x
In this example p
«mple p = q = 1, m = 4, n = s = 2, r = 6, and thence T = 2, w = 1;
therefore the ferics may be fummed algebraically. Subftitute thefe values in the for-
inula -, and M-Ń, and we get
00| 20
X8
2.2+z
+
I
48
for the fum in infinitum; and
J
4
>
for the fummatrix.
3·3+≈
3.6+22
EXAMPLE VIK
43. Required the fum of the feries
?
1.4.7 2.0.9
I
1
I
I
+
+
3.8.11
+ &c.
≈.2+22.5+2%
5
W
Becauſe po, q = 1, m = n = 2, r = 5, s2, it will be r =
سایه
and therefore the algebraic fum of the propofed feries cannot be obtained. But by a
7
proper
OF CONVERGING SERIES.
47
proper ſubſtitution in the formula - (Art. 32.) the fum of the feries will be ex-
preffed by
I
S
* 5:2
I
15.
15
S
x
+
I - X
6
Make √x = y, and this laſt expreffion becomes
2
j
is √ its
S
15
2
5.15
And by putting y = 1 as before, we have
2
2
ys
5
y +
3.15
15
2
L. 2
15
17
450'
the fum in infinitum of the propoſed ſeries.
EXAMPLE VIII.
44. Let the feries propoſed be
I
I
I
+
4. 7. 12
6.10.16 +8. 13. 20 + &c.
I
8+4=
2+2≈.4+3≈
4 +3≈.8 +4≈
It will be p q = 2, m = s = 4, n = 3, r = 8, and therefore 1, = }},
which indicates that the feries cannot be fummed algebraically.
But the values of p, q, &c. being written in the formula P-Q, there arifeth
༢
༢
10
S
I - X'
16
3 S 24
x 4: 3 x
X
3x
16
32
I *
the ſum of the ſeries. And by writing 3 for jy, and taking the fluent, it
becomes
3 L.I
L. 1 − y³ +
3
L. 1-y
16
16
3 L.Ity ty²+
32
9 y
4. 16
+ 9/8/8
ay
16
31, 3
16
6
yo
32
برامه
arch of a circle, rad. 3, and tangent
± √
3y
4+2y
Now put y = 1, and the fum of the propoſed ſeries will be truly expreffed by
31
4. 16
3 L.3-
3
arch of a circle, rad.
√3, and tang. 1.
32
8
EXAMPLE IX.
45. Laftly, let the feries propofed be
I
Ι
I
+
I.2.5 2.3.7 3.4.9
I
&c.
≈. 1+≈. 3+2≈
I
Here
48
THE SUMMATION
Here we find = 0, q = m = n = 1, r = 3, s = 2, and ™ = ½, w = {. Write
thefe values in the formula P-Q, and the fum will be expreffed by
I
S
3
I -x3:2 x x
1+x
:
S1-112 X *
Let x bey, and this expreffion becomes
تو رو
1-x
23
23 S 4 + 4 S 1 - 2 S 22,
3
درو
4
3
L.1+y+ y
4 arch of a circle,
3
And by making y
9
rad. 1, and tang. y.
y²
4
- y − y² +
3.
1, the fum of the feries will be
4
.2 + arch of 60°, rad. 1,
17.
19
3
||
NOTES,
OF CONVERGING SERIES.
49
NOTE S,
EXPLANATORY,
ON SECTION III.
ART. 32.
By writing the value of F in the expreffion G, and again the value of G in the
expreffion H, we get
But
I
小
​S
I
11
ข
I
S S S
S.
n
Sx
xp: 9 x
xm: n−p: q—1 X
Xxm: n~p: q~1 * × x*: s—min—I *, or
n
S
I
xm: n—p: q- X
IFX
I
S
x Pix
I
**
;
9
m
qn X
n
P
9
S
xp: q x
【7
s
xm: n—p:9
x P : q x
X
by Art. 123
+I
in
qn X
72
9
x m : np: 9
and by adding the indices in the latter member, and taking the conftant quantity
from under the fluential fign, there arifes
xm: np: 9
qn X
71
P
S
xp: q x
T
xm: nx
;
F
m
P
IFX
q n x
N
12
9
I 2
which
30
THE S. U M MATI ON
which multiplied by xr, the fluential fign prefixed, and the whole drawn
I
into gives
S
I
am: n−p: 9+r: s−1 ; »—I
x p : 9 x
އ
S
727
Р
17*
q n x
n
9
I
min
I
S
S
ди х
7/1
n
xr: s- -p:9
:9-1
771
Р
S
xm: nx
-
9.
x P : 9
9 x
I
-mn-1 x
xm: nx
nq x
n
Р
9
IFX
S
771
р
17x
nq x
n 9
And each of theſe terms may be again refolved into two fluentials thus,
By Art. 12. we have
but ==
becornes
ท
S
S
"
9
It is alfo plain that
'T
N Q S T X N
X P: q
17x
I
Nqs π X π-
S² 1FX
xp: 9+ = x
>
hence p: q + = = r : s, .. the fluential in the latter member
ris,
And after the fame method is the other exp effion for Q found.
xp: 9 x
I FX
S
X' r: sx
I
is
X x P : 9 x
17x
11
r: s
and in like manner is
for x X I -* ** = xp: q
xP:q+" = xp: q
1
Xxm:&
1FX
Thewn to be equal to
x.m: n
17x
1FX
ART.
33.
I
Απ
By putting
1
x
into a ſeries we manifeftly derive the expreffion P, which from
actually multiplying by xq x becomes
I
nq S T X T
S
» P: 9 x + xp÷9+ix + xp:q+2x + &c. xp: q+~¹÷;
q
and the fluents of the terms under the fign are plainly
xpiq+1 xp:9+2
p:9+1 p:9+2
T
+
Now let cach term be divided by q,
the common factor without the fign,
+ &c.
xp: q+x
P: 9+π
1
fince that value is found in the denominator of
and their denominators become reſpectively,
:
p: 9+1
OF CONVERGING SERIE S.
51
Р
P: 9+1 × 9, P : 1+2 X 9, &c. but 2 + 1 x q = +
9
p+q
x q
x q
9
9
9
29
+
9
=p+9 + 2 x 9
Р
9
9
× q = p + 29, &c. from whence appears the ex-
preffion ". And by a like proceſs is deduced the formula denoted by ; as alſo thoſe
P.
of Art. 34. denoted by P and Q.
ART.
35.
The expreffion
there arifes
I
72
F
n
I
9
S
I — A
A z
xp: 9
x
I
S
S
I
9
S
being written in the formula G (Art. 9.)
I-xx
xP: 9x
X xm:n-p : 1−1 x =
I-X
I
x = X x P : q
xm: n—p : q—1 ÷ X
9
I-X
772
P
write =
w as before,
12
9
Refolve this into two fluentials (Art. 11.) and for
and it becomes
Ι
Xπ
X
219
S
I ༢ ༢
xt: q x
I
1 **
724
S
1
I
Х
I .t
Now, fubftitute this for G in the expreffion H (Art 9.) and there is produced
I
ngs X π-w
I
S.
S
S=
S
1-x
X 1
~ X x p: ? x
I - X
ngXã
But fince - W =
771
11
9
the fum of the indices in the firft member of the ex-
preflion under the exterior flucntial fign becomes.
77
n
In
interior fign
22
nq s X
y
777
2
+
I =
S
72
11
P
1; and in the latter under the
9
#!
therefore the whole formula Lecomes
X X PX
L
S
---
2:48X7-0
-
THE SUMMATION
TION
1
I
ngs X π-w
S
I - X 25
xminx
xris-m
r¦ s~}}} } n—I
} x
- H.
Now let each of thefe members be again refolved into two fluentials, and the formula
arifing will be
ng s
•
I
T
د
r:s-pige
X
S
xris-p:9
-p:9
S
Xxx xp: 9 x
1
X
I - x?
X
nq s
I
•
r:s-p: 9
= X
T
W
r:sm: n
I - X
xP: 9x
· X
ngs
T
W
•
xrism:n
S
xz
•
४.
I
+
ngs
•
ㅠ
​W
X
S
xris-m : n
X
T: J-m: n
• x min
;
x-I
but
S
9
xr:54:9
Nqs π X π
w, hence the expreffion hec mes
+
m
π
and
= w,
S
n
S
xp: 9x
1-x
ngs
S
xx.xm:n x
+
I
xris-m: n
ng sw X π-w
I
Sπ X π
I
n q s w x π --- W
S
S
I
Xx
xr: 5x
I - X
xz
xris x
;
I - X
and making in the finite quantities, we derive the latter expreffion denoted
by H.
* = 1
And the former part of this expreffion is plainly equal to
I
1
•
xx xp: 9x
X
Nqs π
T
W
I-X
•
xx
xris
•
I-
X
and because the fluent of the difference of two quantities is expreffed by the difference
of their fluents, it becomes
I
S
X
I ·xx XxP: 9 x
xris x
; but
nq s n
SL
I X
xp: q
* ris
= 1
P
4
= 1
and .'. I—x* X xp: 9 = xpig --- xris,
xp: q
hence this part of the formula becomes
I
T
I
+ z
And in like manner does
appear to be equals.
S
X Z
•
I
Π
×π . x Þ : 9 x
X
xx.xm:n x
1-X
*
xm: n
S
I -
The
OF CONVERGING SERIE S.
53
The expreffion M is transformed into M thus,
**
Multiply 1 by 1- and the product is
x
π
which divided by 1-x the quotient arifing is
I
X x* + x≈+”,
1 + x + x² + &c. + xπ—1
Xz
-x²+1- &c.
x²+π−1 ;
and this multiplied by x:9 produces M. By the fame procefs is Ń found.
Laſtly, the fluents of the terms of M are plainly
xp:9+x+2
&c. -
xpq+z+1
1:9+π p:9+x+1 P:9+2+2
xp:q+1
P:9+1
+
xP:9+2
P:9+2
xp:q+=
+ &c. +
xp:9+x+x
7
p: 9+x+=
9xp:9+1
=
+
p + q
9xp:9+2
p+29
+ &c.
9xp=q+x+=
9xp=q+x+2
qxp:9+x
qxp:9+2+1
p+xq p+q.z+1 p+q.x+2
&c.
+
p+q·x+x
and by making × = 1, and bringing the terms to order, they become
9
q
P+9.2+1
Z
+
9
p+29
p + q
or by reducing each refidual to a common denominator
99.z
p + q x p + q.2+1
9
p+9.2+2
+ &c.
9
q
p + = q
p+9.2+=
972
+ &c.
992
P + 2 q xp +4.2+2
which drawn into the common factor
I
11 9 Sπ- 7-
p + = q x p + q. Z + =
gives the value denoted by
"
M. In the fame manner is the latter member of the formula (N) inveſtigated.
And by a fimilar method may the formula M-Ń (Art. 36.) be derived from the
general fummatrix A, without fubftitution in the laſt formula M—Ń.
ART. 37%
N.
From comparing the particular or numeral denominators with the general or literal
one we have the values of the fymbols p, q, &c. as in the example, and thence
y
P
3
777
S
9
I
I
3 = *, and
3
I
=w; we mußt
$
n
I
I
therefore
7
54
THE SUMMATION
(Art. 33.) for the fum of the feries.;
therefore take three of P terms, and two of
///
"/
as alfo the fame number of M and N terms refpectively (Art. 34.) for the fum-
matrix.
The former gives
(P)
I
I
I
I
I
I I
I I
X :
+
+
X
IXIX3X3-2
0+ 1
0+ 2
0+3
3
6
18
(ë)
I
I
I
I
5
x:
X
2
6
12
I I
18
(M)
(Ń)
IXIX2X3-2
5
12
Z
-
1 XIX 3
Z
I XIX 2
7
36
+ (
I+I 1+2
the fum ad infinitum; and the latter produces
IXZ+I 2xx+2
I
I
I
X:
+
+
3Xx+3
I
X:
I
I+I X 1+z+I
+
!
}
Z
I
I
I
Z
X :
+
+
3
Z+I
2.2+2
3.x+3
2
א
= = G
X:
x+1
2+
2
2
+
+
=
2|0
Z
2
X:
alt
2.2+2
2
3.2+3
I + 2 × 1+x+2
22/0
-: X
X
I.
2.2+2
3
.I
3.2+3
3
+
2.2+2 3.2+3
3
2
3
+
+
2.2+2
2.2+2
3.2+3
3.≈+3
א
Z
2
I
I
www
X:
6
Z+1
2.2+2
as in the example.
To exemplify the fummatrix, let the ſum of 60 terms of this ſeries be required.
Here then z = 60, and ..
3.2+3
2
IO X :
61
I
I
124 189
138895
714798
is the required fum.
A different method of inveſtigating the fum of this feries may be ſeen in De
Moivre's Miſcellanea Analytica, p. 115.
ART.
38.
2, and I, hence
Here 2,
I
+ (= 5) - (()
T
(2)
I
X :
18
I
5
I
72
36
24
4
I
-1
72
the fum.
I
I
X :
(=
1+3
136)
And
A
OF CONVERGING SERIES.
55
]]
And (M) × :
210 210
I
Ι
+
IX 32+1
4 X 3x+4
"/ Z
(N) X
3
I
4 X 3≈ +4
I
I
2
X :
6
+
3x+1
4·3%+4
4·3%+4
א
I
X :
I
the fummatrix.
6
32+1
4·3%+4
The fummation of this feries is given by Stirling in his Tra&. de fumm. Ser. pag. 25.
Examples 3, 4, 5, and 6, are applied exactly as thofe preceding, and therefore
need no particular illuftration. But with refpect to the first of thefe, we cannot but
obferve, how much more elegant and concife this method of fummation is than that
given by De Moivre in his Mifcell. Analyt. p. 114. for the fame ferics. And the two
latter, it is worth notice, are of thoſe ſpecies of feries which have been hitherto deemed
incapable of fummation; the factors in the denominators of the terms being neither
immediately confequent to each other, nor belonging to the fame arithmetical pro-
greffion.
A R T. 43.
Here T =
5, and w =
therefore the feries does not admit of an algebraic
2
2
ſummation. But by writing the values of p, q, &c. in the formula –ά (Art. 32.)
we have
s
I
ΙΟ
S
xx-x
-x x
I
I-X
= S
I
ΙΟ
S
I
3
X
x x x
I
S
X x x x
I-X
I
لم
10.
I X
+
6.
I
ΙΟ
S
I
+ SS
I
15 1-X -6
I
6
but S
10 -
S:
x x
I-X
-
I
x x
S
+ xx
I
;
I-
*
ΙΟ
I
I-X
I - X
I
-
訂
​I
I
10
I-
S
x x
x x
X
1 - X
I
S
I
6
- ÷ S # = S ÷ +
6
I
Si, for
= 1 - ÷ ; S
6
15
S₁** = √ *=**** = S₁₂- S
I-X
I
*; hence the fluential becomes
Six-Si
I
K
I
15
56
THE SUMMATION
I
S
λ 5:2
2 x
15
S
x
+
I X
6
15.
I-X
Now, the fluent of the two latter terms of
this expreffion, it is obvious, may be had without any farther reduction; but the
index of the firſt term being an improper fraction, denotes it to be of that ſpecies of
Aluxionary expreffions, of which the fluents are not explicable in finite terms, without
a transformation. We muft then fubftitute fuch a quantity for x, as will make the
exponent become either a proper fraction, or (which will give a much neater fluent)
a whole number. Writing therefore y for x, we have xy, x²=y, and x =
2yj; hence the expreffion becomes
yó
y²
but S223
S₂
I
6
yº ÿ I
y j
1 S22 - 1 S 222 + 2;
15
J
y
-S223
2
15
S
may be transformed to 2-S28 3 – S23'3-
2
2; for the first member SAYYE
4
6
2 y* j − 2 y¹ j + 2 yº ÿ
Szyty
y'
S223 -Say's
I -y
2 j
4
2 j;
I
2
is plainly =
2 34 j
I
ل
≥ yª ÿ — 2 yº ÿ
I-2
y² — y² j yª
and S 22') is = ƒ 24's=212 +27 3
4
2 j y¹ j
S²²²² - S²rs
z y² j
2
2
=S
+ 2 y 3 = S2-S 27's and again,
— y²
2
'2 j − 2 j + 2 y³j
2
I
2 j
ل
y
S₁22, -S2;; laftly S, 23, -S22,
I — y²
above fluential) is =
I y²
I
2 ÿ — 2 y² ŷ
(the latter member of the
2ÿ — 2 yÿ = (dividing the numerator and denominator by
-3)23, confequently by collecting the terms we have
y
S 223 - S 223 = S 223, -S 25 ; -S253-S 23.
=S
y
Hence the fluent is X: hyp. log. 1+y
درو
نو 29
4
j
2
303
y+
+ 2/2 2,
1.5
5
3
or, when y = 1,
2 L.2
2
2
2
I
17
+
+
=====
||
L. 2
15
5. 15
3.15 15
6
15
450
4
We
OF CONVERGING
57
SERIES.
We may often transform a fluxionary expreflon into a more convenient one, by
throwing the parts of it into a ferles; and then by adding, fubtracting, &c. the
terms arifing, according to their figns, we may difcover fome finite quantity equal to
the refulting feries, being redundant or defective by one or more fimple terms, which
must therefore be connected to the finite quantity with their proper figns, So the
expreffion in this example may be transformed into a more fimple one thus,
1Ο
12
I — y²) 2 y°} (2 y^ j + 2 y³ j + 2 y¹º° j + 2 y¹²j + &c,
I
1 — y³) 2 y ỳ (2 y j + 2 y³ ÿ + 2 y³ j + z y' j + 2 y° j + 2 y¹¹j+2 y¹³ j + &c,
Subtract the lower feries from the upper, and there remains
7
— 2 y ý — 2 y³ j — 2 y³ j + 2 y° j— 2 y¹ j + 2 y³ j − 2 y° j + 2 y¹º°j
&c.
Now, if we fupply the terms that are wanting to make this feries regular, namely,
2 j + 2 y² j + ? yj, it becomes
4
2 j − 2 j' j + 2 y² j − 2 y³ j + 2 y¹ j — 2 y ‹ j + 2 y° j — 2 y¹ j + 2 y³ j &c.
2 j
I+Y
which is manifeftly = expreffed in a feries. Confequently
2 y→
I-y² 1
2 y.
2)
2 j − 2 y³ j — 2 y¹y, as before,
I+Y
AR T. 44
By fubftituting the values of p, q, &c. in the formula - (Art. 32.) wo
have
I
3.2.4.1 X 1
s
I-X X X
I va X
3
୪
+ S - S t +
3
$6
**
I
S
*2 *
I-X
I
3 · 2.4. XI
; but S
(by the fame proceſs as in the preceding Art,)
x x
3 S + + - 3 S * *, and again
I
*
160
the expreffion becomes
3
16 I
3
*** is
S
3 S - - - S - 3 S +
I
ex, that **
16 1
मे
16
3x
-
16
32
3
16.
S
*
52/00:
X
S
x + + +
Six-
is=
I
X
IX
11
as in the example.
I - X
3
=
that →³ may become yª, having an integral exponent, then is x ≈ y³,
3y²; theſe values written in the laſt expreffion produce
Put y = x
and
✰
K 2
IÓ
010
58
THE SUMMATION
ل
y²
35
3y3
6
3 S 323 - 3 S 323 - 202 - 12.
16. I
16. I
16
32
Now, the fluent of the firſt term is — 3 hyp. log. 1—y³, by Form 4th, Emerſon's
16
Fluxions; but the fecond term not admitting of a finite fluent may be reduced into
16
9
3 j
3 16
9
16
- 3 S 2 - 3 S F + - 3 S › = 36 S ²²
3y
914
j
+
+
4.16 16'
9y
y
for
تو رو
3
is manifeftly =
تو دو
y3j, and
تو درو
3
j.
And the fluent of 3ÿ
may be had by Form 8th, as follows;
By comparing the fluxions we have
a = + 1, B =
180°
I, ≈ = y, " =+ 3, ♪
n = + 3, d = 1, λ = 3, *
3
x =
= ور =
=y, K =
= 60º, a = fine of 120° = + v² (rad. 1.) scofine of 120°
3
J
እ
And becauſe =, we have by Cafe I. A =
I, B =
25 = +1, C = 2 a
= √3. Theſe values being written in the general fluent, produce × : − L. 1−y
+ L. Vi+y+y² + √3 × arch of a circle, fine
I
vay
4
3
= (by note to Art.
I
30.) X : - L. 1−y + { L. 1+y+y+ 2 × arch of a circle, rad. ½ √3, tang.
1
1-
3y
4+2y
; which multiplied by
16
3-3, and the figns changed (3S 25,
being ne-
3
gative) becomes
32
+ 3 L. 1-y- 3 L. Ityty - 3 circ. arc. rad. √3, tang.
3y
4+2y'
as in the
,
16
example.
As fluxions of this kind are the most difficult in their application to the general
formulæ for the fluents, we ſhall again fhew the inveſtigation of the fame fluent by
Simpſon's method.
The fluxion correſponding is that in Prob. V. p. 371. Flux. where by comparing
the forms
xm-1 x
rn xn
تو
ڈرو
I
we have m 1, n = 3, r = 1,
x
x = y; b = cofine of o° = 1, 6
I, c = cof. of
360°
m
(= 120) = -, rad. 1. R =
3
× 360° = 120°, fine of R = + √3,
n
cofine
OF CONVERGING SERIES.
59
ά=
cofine of R= -1; Q = 0, Q = 120°. Hence Mhyp. log. √1—26x + xx
M
b
hyp. log. 1—y, M = hyp. log. ✅1−2cx+xx = hyp. log. √ï+y+y². From
whence N = 0, Ń = arch of a circle, rad. 1,
fine
é
* X fine ά
x
√r r-2 crx + xx
- circ. arc.
fine
1+y+y`
21
And by Corol. p. 372, the fluent
rm.
n
X : M + fine R × 2 Ń
cofine R X 2 M,
by fubftituting the above values, becomes
X
hyp. log. 1-y+ x 2 circ. arc. rad. 1. fine
√2
+플 ​X 2
Vi+y+ y²
hyp. log.
1+y+y²; which may be reduced as before.
The fame investigated by Landen's Theorems, p. 99. Lucub.
The expreffions compared are,
xm-
I
نو
an- X"
from whence a = 1, m = 1, n = 3, and therefore c = tang.
ес
3
180°
60°
(rad. 1.) ≤3, A = circ. arc. rad. 1, tang. √3, M = 2 × fine of 2 A = (120°)
2 √ 2 = + √ 3, Ń = 2 × cofine of 2 A = I, x = y; and the fluent
am
12
× : M (CPD) – (DP: CP) - Ń (DP: cé)
becomes
X:
+ √3, arc. (CPD)
(CPD) — hyp. log.
DP
CP + hyp, log.
DP
CP
Now, to determine the values of the meaſure of the angle CPD, and of the fractions
DP DP
CP,
cé
I
, we muſt obſerve that (Fig. 2.) Pá, áÁ, &c. = — × PO = † × 90°
12
= 30°,
60
THE SUMMATION
= 30°, •.• PÁ = 60”, and PP =
3, CP being 1. And from the theor, we have
BP: CD :: @+*: that is 1+yly ¦¦ ¦
*; @
DP
DP
ay
CP = 1/3 DF = (³D²+PP |* =) √3 +
of
3+
=
CD, and DP; henea
Į = 2x y²
2
+3
21 + £¸ C$ = ( PC² + P P²
差
​СР
DP
== ) 2, and therefore
CP
VI+ y + y²
log.
1+ y
11
HIN
3 +
Į→ 21 + y²
1+ 2y + y²
'I + y + y²
I+Y
; but the hyp
ᎠᏢ
hyp. log. Vi+y+y² → hyp, log. 1+), and hyp. log, CF
hyp, log. 1−y + hyp, log. i+y.
Moreover, PD: DC :: fin. DCP : fin. CPD, that is
√4 +49 +41 2y
I+y
I+y
And collecting the terms, the fluent will be
Nay
√x + y + y²
hyp. log. I−y + √3 × circ. arc. rad. L
fine
+ hyp. log,
I + y + y²
√i+y+y², as before.
ART. 45.
If *=y, we have x = y², x*
we obtain
*
تو بیو X -درو
- SFX -
WIN WIN
23 S 4 + 4
++
The fluent of the firſt term (+
≈ y³, and * = 2y); whence by ſubſtitution
2
SEX.
-y × y³}
نور
S 1 - 2 SB.
Fluxions). And the fecond term
у ў
y²
) is † hyp, log, 1+7², by Form 4. (Emerfon's
I + y²
2
Info y'
) is reducible to
afr
1+y²
j
·y²
4
y
for f
2)² = 1 ) + 1 j + 2 2 == 2 + (2222); },
}
I+y
y²
+ y²
14.
ولا و
and
OF CONVERGING SERIE S.
61
نو شو
and
1+ y²
2
−j+j+ÿ² ÿ
1+ y²
تو
= + 4,- (+))
· j + y²
į
) ;;
j
alſo the fluent of
1+ y²
is circ. arc. rad. 1. tang. y, by Form 5. confeq. the
fluent of the ſecond term is
+ circ. arc. rad. 1. tang, y, + y³ — y.
تو درو
تو لو
The third term (-
2
I + y²
) is alſo reducible to +1: -1; ;
y²
تو درو
−ÿÿ + ÿ ÿ+y³ÿ
3
yj
for
= +
2
I+y
I
1+ y²
I+ y
y j + y³
i+y²
the fluent of which is
+
hyp. log. 1+y²
플 ​yo
نولو
The fluents being collected, and multiplied by their refpective coefficients, we
have
÷ × { L. I + y + circ. arc. rad. 1, tang. y, + × y³ − y + 2 × L. i+y
44
1 {
1
2 × y² = † L, ¡+y+ ŝ circ, arc, rad. 1, tang. y, + § y³ – ŝĝy — y².
SECTION IV.
62
THE SUMMATION
1
SECTION
}
P
RO P. IX.
IV.
i
}
46. To investigate the Sum of a Series, when the Denominators of the Terms confiſt of
four fimple Factors, being generally expreſſed in this Form
p + q.m + n
I
m+n.r+s. t + u
I
+ &c.
p+ 29.m+2n.r+zs. + + 24
I
PTY Z. M Tux.r+sz. t +uz
The formula for the fum of this feries is
I
I =
น
s
Hxt: u-rs-1x, Art. 9.
in which let the values of H, G, and F, be fubftitured in order, and the refulting
expreffions reduced according to Art. 11.
t
P
9
t
t
m
w for
d for
,
22
น
12
π
and we fhall have, by writing for
and making = 1 without the fluential
w
u
fign,
I =
π n g s u
I
x p : q x
I
T
W. T
IF*
πnqsu •
- W
π
I
I
+
u ný s u
น.
- W
Π
sngsu. s
su.d-w
ㄠ
​J
π
+
w nqs u.
47. Now, the firft fluential contracted is
I
Engsu.
su.d-
I
w nqs u. d-w
S
IFX
xm: n.x
w n q s u
π
•
+
T
xr: 5 x
T
I
πngsu. π-W ω
T
T
s
I
яп xp: 9 x
•
IFX
S
S
X 1: u x
IFX
x 1:1 x
17x
xli u x
A
1 : u
IFX
the
{
OF CONVERGING SERIES.
63
the third
I
S
x.xm: nx
IFX
S
-- X4
x r : s x
1FX
dnqsu.d—w.π-
and the ſecond and fourth become
I
w nqs u
•
Conſequently the formula I is transformed to
I
(R)
π nqs u
•
π W T
-xs
• xm : nx
+
IFX
S=
I
w n q s u T
•
xp: 9x
IFX
S
I
I
X
Duqsu.
*
x
IFX
Hence if the propoſed ſeries be affected with poſitive figns, the fum may be expreffed
algebraically whenever , w, ♪ are whole numbers; but if the figns be alternately +
and, the fummation cannot be fo exhibited, unleſs, w, and ♪ be even whole
numbers. For when the figns are affirmative we have
Alfo
And
I — X™ xp: q x
9 X
S
n x
I - X
xp: q+1
p+q
+
xm: n
1 -- X
x #: n + 1
m+n
لدير
x ritx
=
S
I+x+x²+x³+x++ &c. x-I X xP:9; =
sp:9+2 xp:9+3
+
4+24 P+39
S
+ &c.
xp:q+x
p+= 9
↓ $+*+*²+*²+**+ &c. x
m+2n
xminta
+
+
+ &c.
m+3 n
mtôn
s
xriste
*
+
+
r+s
I - X
* isti
[+x+x²+x²+x++ &c. *"ti
r+25 r+35
+ &c.
xristw
x+45
Making therefore ≈≈ 1, the fum will be truly expreffed by
(Ŕ)
I
T
ofa
p+q p+29
Xx
I
Xx
I
X
*
ris
IT
I
I
X:
+ &c.
P+39
p + = 9
I
I
I
I
I
1
X:
+
+
+ &c.
sqsu.
d 9
m+n m+2n
m+31
m+dn
I
+
X :
wn qu
where as many terms of each member muſt be taken as there are units in
L
๔)
♪ and a's
refpectively.
I
I
+
+
+25 r+35
+ &c.
1+05
64
THE SUM
SUMMATION
reſpectively. But if the figns of the propoſed ſeries be alternately changeable from
poſitive to negative, and the exponents x, d, w, be even whole numbers, then the
expreffions
I -xπ • xp: 9 x
I + x
become reſpectively
S=
x min x
I+x
S
I
: *
I+x
xp:q+1
I
xp:9+2
xp: 9+3
9 X :
+
&c.
p+q
p+29 P+39
p+q
xm:n+z xm:n+3
xmintð
nx :
+
&c.
m+n
m+2n m+3n
m+ & n
SX:
r+s
r+25
xr:s+2 xris+3
r+35
'
xristw
+
&c.
r+ws
And, writing 1 for x as before, the general expreffion for the fum of this ſeries
will be
I
I
I
I
I
(K)
πnsu.
X:
+
&c.
⇓) ---- L
T
p+q
p+29
p+39
p + π q
I
I
I
I
I
X:
+
&c.
8qsu. d-w.π
m+n
m+ 2n
m+3n
б
m + d n
I
I
I
I
T
+
X:
+
&c.
angu
IT
ω
W
rts
r+25
r+3s
П
as many terms being taken as there are units in , d, w, respectively.
r+ws
For if in the
—, then ex-
**,
48. The general fummatrices of theſe feries are easily inveſtigated.
formula R, we multiply each member under the fluential fign by
pand the terms by dividing by 1-x, and reduce the whole by Art. 11. there will
arife a general expreffion, the fluent of which being taken, and in every tetm x made
=1, the refult will be
(Ś)
9%
I
I
X:
+
+ &c.
πnsu.
T
T
d
I
T
p+q.p+q.z+1
p + π q⋅ p + q.x + x
P + 28. p + q.x+2
NZ
sqsu.
X:
I
m+n.m+n.z+I
· I
m + d n . m + nix+8
+
I
+ &c.
m+2n.m+n.x+2
of
+
Q F CONVERGING SERIE s.
65
+
X
wnqu T - W
I
r+sorts.%+1
Ι
rts. xtar
r+ws.arts
+
+ &c.
r+2s.r+s.x + 2
I
وه
the general fummatrix for ſeries of this form affected with affirmative figns; as many
terms being taken in each member of the formula as there are units in ≈, d, and
reſpectively. And by the faine method may the general fummatrix be found when
the figns of the propofed feries change alternately from poſitive to negative, only here
we muſt multiply the flowing parts of the formula R by 1-x2%, and divide by 1+x.
But the fame thing may be effected by only writing 2% for z in the laft expreffion (Ś),
and making the figns + and - alternately; for then it becomes
292
(ő)
πnsu. F-W
}
I
X :
+ &c.
p + q⋅ p + q.2+ &
p +29⋅ p + q.2% +2
I
2nz
p+πq · p + q • 2z+x
su.d-w.π
8 q su
•
X:
б
I
I
+ &c.
m+n.m+n.2z+1 m+2n.m+ñ.28+2
I
+
25%
wnqu. π-w.
m+dn.m+n.2z+d
X:
W
I
I
+ &c.
rts.rts. 2z+ I r+2s.r+s.2%+2
I
r+ws.rts. 22+ w
;
T
or initial differences, of the
as many terms being here alfo taken as there are units in , &, and w, reſpectively;
which then exhibits the fum of as many pairs of terms,
propofed feries, as there are units in z.
EXAMPLE I.
49. Let the feries propofed be
I
+
I
+
1.2.3.4 2 · 3 · 4 · 5
From the general term we obtain p
t=3.
Whence ™ 3, * = 2, w = 1,
I
+ &c.
25
I
3 · 4 · 5 ·
1+2.2 + 3.3+2
= 0, 4 = m = n = √ = # =], /= 2,
I
and therefore the' feries may be fummed
L 2
algebraically.
66
THE SUMMATION
algebraically. For ſubſtituting theſe values in the formula Ś, the fummatrix arifing
will be
Z
I
I
I
X:
+
2.3
1+ 2
2+% 3·3+%
And the fame values of p, q, &c. being written in the formula Ŕ, give the fum
in infinitum.
EXAMPLE II.
50. Required the fum of the infinite feries
I
18'
I
+
2.4.5.6
I
+
I
3.5.6.7 4.6.7. . 8
+ &c.
1
1+2.3+%.4+%.5+%
Here p q = n = s = u = 1, m = 3, r = 4, t = 5;
= 9
♪ = 2, w = I. And becauſe the fum
3, r = 4, t = 5; and therefore π = 4,
of the feries may be had algebraically, let
theſe values be written in the formula S and Ŕ; and the former will produce
Z
1
I
X:
24
+
•
2 2+z
3·3+≈
5
4.4+%
3
+
5.5+%
the fummatrix; and the latter
I
I
the fum.
3.5.9
135
III.
EXAMPLE
51. Let the feries propofed be
I
I
I
I
+
2.6.4.5
4.
8.5 6
+
+ &c.
Here we have p = 0, q
6.10.6.7
2%.4+2x·3+≈•4+≈
and = 4,
= n = 2, m = t = 4, r
n = 2, m = t = 4, r = 3, s = u = 1; and ☛
♪ = 2, w = 1. Thefe values fubftituted in the formulæ S and R produce, re-
ſpectively,
Z
X:
24
I
I
5
3
+
+
>
4.1+%
8.2+%
12.3+%
16.4+%
7
7
the fummatrix; and
=
the fum.
>
22 32
1152
E X
AMPLE IV.
52. Let the feries propofed to be fummed be
1
1.6.10.21
I
I
1
+
&c.
2.8.12.24
3. 10. 14. 27
≈.4+2%.8+2z. 18+3%
By
OF CONVERGING
67
SERIES.
By comparing the terms we have p = 0, q = 1, m = 4, n = s = 2, r = 8,
= 6, ♪ = 4, w = 2. Which values being written
t = 18, u = 3, and therefore
in the formula S and Ŕ, we have
Z
I
I
I
I
I
T
X:
+
+
12. 24
I+2%
2.2+2%
3.3+22 2.4+2%
6.6+2%
5.5+2%
the fummatrix; and the fum in infinitum =
4223
6635520
EXAMPLE
V.
53. The feries propoſed is
I
I
I
+
+
+ &c.
Z
1.2.3.4 2.3.5.7 3.4.7.10
I
1+z.1+2% . 1 + 3z
In this example, p = 0, q = m = n = r = t = 1, s = 2, u = 3, hence ==
I
2
I
d=
W
>
; and therefore the fum of this feries cannot be exhibited
3
3
6
w
in a pure algebraic expreffion,,, and a being fractional numbers. We muft
therefore have recourfe to the oecumenical formula R (Art. 47.) in which ſubſti-
tuting the above values of p, q, &c. there arifes.
x 1:3
1 : 3. *
I-X
S
- x −2 : 3 . x x
+
SE
expreffing the fum of the given feries. Now, put x
1:6 -
ولا
-1:6¸ x 1: 2 ✰
•
I-X
that the exponents of
the flowing quantity may become integral, and the laft expreffion is changed to
6
s
I-y vy
I-y
6
which by a proper reduction beconies
I—y
¡ . yˆ j
+24
S
7
•
;
Si
уў
تو
123' - 8y + y² + 24 S 1 2 3
2 - 24 f 217
y²
I-y
1
1 2 y² — 8 y³ + y + 4 L. 1+y −4 L. 1-y-8 L. y + +++L. 1−y²+8 circ.
arc. rad. ✔3, tang. y. Correct the fluent, and make y = 1; and the fum
of the propoſed infinite feries will be truly expreffſed by
§
5+4L.2−8 L. 3-8 circ. arc. rad. 3, tang. 1.
EXAMPLE VI.
54. Laftly, let the feries be
I
I
I
I
2.3.5.7
3. 5. 8. 11
+
&c.
4.7. II. 15
1+2.1+22.2+3≈.3+4%
It
68
N
THE SUMMATI
It will be p = q =
4
4 S
ω =
m = !, n=r=A, S
3, $= 3, ❝ = 4, and hence &
12
; the feries therefore cannot be fummed algebraically. But
by writing the values of p, q, &c, in the oecumenical formula R, the fum of the
given feries will be expreffed by
·
TI: 4 x3=4x
+ 2
1+x
I -x 1:4. x
1+x
xx: A X
I
•
AI: 12 x²:3 x
9
It a
To reduce the exponents of the flowing quantity into integers, put *1:12 ➡y, and
the above expreffion becomes
•
12
S
y²° ÿ
I+Y
1-79) + 24
S
==
—y³ •
1+12
y
12
19
"- 108
y • y
)
The fluent of which being taken, and properly corrected, gives
3
2
[2
hyp. log 1+y'
HIN
8 ✔ž hyp. log. y³ — √//}}} + { 8 hyp. log. y³ + √ ½ + 3/1/20
1 2
16 circ. arc.
ག
rad., tang.
2 y³
2 yo I'
4 circ. arc. rad. 1, tang. yº,
$
+ ½ hyp. log.y²+
3 y
— 9 hyp, log. 1+y² + 18 circ. are. rad. ✔3, tang. 2y++++; or by making
I
4 4
} = 1,
I
2-1/2
8 √ ‡ L.
8 L. 216
16
6
circ. arc. rad. √, tang, 2, -- circ, are,
2 + √ 2
180º, rad. 1, + 18 circ. arc, rad. {√3, tang. §; which is the fum of the propofed
feries ad infinitum.
1
NOTES,
OF CONVERGING SERIES.
69
N
T
E S,
EXPLANATORY,
ON SECTION IV.
AR T. 47, 48.
The general formulæ denoted by R, Á, Ŕ, in this ſection, are derived exactly as
thoſe in the preceding one, namely, R as P—Q in Art. 32. Ŕ as Þ – Ő in Art.
33. and Ŕ as ❞— in Art. 34. And by the fame process as the formulæ M-Ń,
P Ő
"
M–N were inveſtigated (Art. 35, 36.) may έ, and S be deduced. For when the
figns of the propofed feries are pofitive, the first principal fummatrix
I
q
S
—
x≈. xp : 9 x
I
.t
(Art. 10.) being written for F in the expreffion G (Art. 9.),
and this value of G again written in H, and laftly this value of H fubſtituted in I, we
ſhall derive the expreffion R drawn into I-x, from whence we proceed as in the
inveſtigation of the formula M-Ń (fee note on Art. 35.).
And when the figns of
the given feries are pofitive and negative alternately, by writing the ſecond principal
J
fummatrix
I x2x xp: 9x
•
q
1++
(Art. 10.) for F in the formula G, and again
the formula G in H, and laftly H in I as before, we ſhall have the formula R drawn
into
70
THE SUMMATION
into 1
tigate S.
2, from whence we may proceed as in note on Art. 35. to invef-
It is obfervable from hence, that the formula M-N might alſo have been imme-
diately deduced from multiplying P-Q by —, in the fame manner as that de-
I
noted by R is produced by multiplying R by 1-x.
ART. 49.
The general terms compared are
I
Z. 1+2.2+% • 3+z
from whence appear the values of p, q, &c.
p + qz. m + nz
I
r+sz.t+uz
And becauſe the figns of the propoſed
feries are pofitive, and π, ♪, w, are whole numbers, the formulæ correfponding are
Ŕ (Art. 47.) and Ś (Art. 48.).
expreffion, produce
I
3.1.1. I. 3−1 · 3—2
The above values being ſubſtituted in the former
T
I
X: + +
(to (3) terms)
I
2
3
I
I
Ι
-> X:
+
(to ♪ (2) terms)...
2
2
3
I
I
+
(tow (1) terms), the fum;
2
2+1
which reduced becomes
II
5
I
36
+
12 6
Which agrees with De Moivre's Miſcell. Analyt. p. 120.
The fame values fubftituted in the latter formula Ś, give
Z
I
I
I
X:
+
+
(to π terms)
3.2
2+1 2.x+2
3.2+3
I
I
X
+
d
(to terms)
2
2.1+2+1
3·1+2+2
+ 2 = x =
I
(to W
terms), the fummatrix for z
3·2+x+1
terms of the propoſed ſeries; which by reduction becomes
2
I
I
I
X:
2.3
Z+1
+
2+2 3.2+3
as in the example.
Το
OF CONVERGING SERIES.
To illuftrate the formula Ś, let the fum of 10000 terms of this feries be required,
Here z 10000, and the fummatrix becomes
10000
I
I
I
X :
+
6
10001
10002
30009
1000600110000
18010801980108
ART. 50, 51, 52.
Theſe need no particular illuftration, being applied exactly as the preceding one,
the two former being fummed by the formulæ Ś, and K, the figns being poſitive;
"
"
and the latter by the formula S, and K, the figns being alternately + and -. The
two latter it is obfervable, are of that order of feries which have been hitherto thought
incapable of fummation; the factors in the denominators of the terms not being in
the fame arithmetical progreffion.
ART.
53.
Here,, and w, are fractional numbers, which cannot therefore be applied to the
formulæ Ś, Ŕ; we muſt then have recourſe to the cecumenical formula R in Art. 47,
from whence the other formulæ are derived. In which by fubftitution, and proper
reduction (as in note on Art. 43.), we obtain
the fluent of which is
24 Xyj-y² j + \ y³ j +
توو
1
У ў
・y
6
y² —
S
تو رو
123` — 8y' + y² + 24 f ) - 24 S 22.
I
Now the two laft terms may be reduced by either Emerfon's Flux. Form 8th, Simp-
fon's Flux. Prob. V. p. 371, or Landen's Lucub. theor. 13, p. 99, as in note on
Art. 44.
And by a farther reduction of the fine, rad. 1, to the tang. rad.
in note on Art. 30.) in the reſulting fluent, we have
4 hyp. log. 1+y³
I
4 hyp. log. — — 8 hyp. log. y + žľ² + å
+ I
4 hyp. log. -+ 8 circ. arc.
i −y³
I
rad. 3, tang. y+.
3 (as
Make yo, and the fluent becomes 8 circ. arc. rad. 3, tang. , hence this
part of the fluent being corrected, and y put = 1 in the whole fluent, there arifes
12-8+1+4 L. 2-8 L. 3 +8 circ. arc. rad.
2
3, tang.
8 circ. arc.
rad. 3, tang. = 5 + 4 L. 2-8 L. 3+ 8 circ. arc. rad. 3. tang. — 8
✔3,
38 ✔3. 1
arc. of the fame circle, tang. = 5 + 4 L. 2 - 8L. 3 + 8 circ. arc. rad. § √3,
tang.
M
64
THE SUMMATION
tang.
With rad.
5+ 4L. 28 L. 3+ arc. 240°, rad. v. For let AB 1 (Fig. 3.)
take BL, , and draw LD, DC, reſpectively parallel to AB, BG.
BC, defcribe CL, produce CD till DE be 2DC, and join BE.
2 ༡
=
Then is CD
= ½, CE = 4, and CB (= √BD²-DC) = ; hence BC²+CE² = BE
=√3, and... CD: DE:: CB: BE, confeq. the angle CBE is bifected by BD
(Elem. VI. 3.) and the arches CI, IF, equal. From whence it is obvious that +
arc. CF arc. CI
+ arc. CI, as in the laſt member of the fluent.
arc. CI = + arc. IF
I
29
= 30°,
And when AB = 1, and CD = the arc. AD 30°, and theref. CI
which multiplied by 8 becomes arc. 240°, to rad. ž√3 (BC), or tang. ½ (CD).
The expreffion 12
I+Y
ART.
+24
1 + y²²
•
4S.
2
54.
y² 3 y
1+ y
12
•
I+y"
2
[7
19
I-y.y
I+y"
12
may be reduced as follows,
20
ÿ
I + y²
12
I—y³ . y²º ÿ
—
12
yº ÿ + y² ÿ + y²° ÿ
I ty
I 2
=
1+ y'
8
y y
+ y³ j;
8
I+y"
TI
= +
y¹¹ ja
23
y¹¹ ÿ — y¹¹ ÿ — y²³ j
j
12
The firft term
but
and
y² 3 ÿ
1+ y²²
23 y
+
I —y³ . y¹7 j
1+212
17
The fecond term
but +
نو و
تو
I 2
20
23 = 13+ y² 5, and 2
12
12
Ι
I+y'
20
320 j
-帶
​i
12
= +
(2
8
نوو
I+y'
1+ y¹2
17
تو دارو
=
I+Y
And the laſt term
12
— y³ j.
I
I—y. y' 'ÿ
1+y"
T9
19
12
1+ y¹²
12
яго ў
I + y¹²
تو دو
تو رو
12
y' j +
L+ y
I + y²
12
— y³ j.
8
The terms being collected,
fimple quantities y be made
I I
12 ylj
1+ y'
each multiplied by its coefficient, and in the fluents of the
1, we have
24ys i
96 ys j
108 y' j
7
I
+
12
12
12
+ 3/1/15
I+}
I
The Auents of theſe terms may be had by Emerfon's forms, as follows: Here n the
index of the flowing quantity in the denominators is equal 12; and therefore in the
firſt term, 11 = n—1, which belongs to the 4th form, hence the fluent is
hyp. log. 1-y¹².
In the fecond term, 5=n-1, which correſponds with the 5th form, from whence
the fluent is
4 circ. arc. rad. 1, nat. tang. yº.
In
OF
65
CONVERGING
SERIES.
In the third term, 81, which is the 7th form, cafe 4th, hence the
fluent is
8 × 2 hyp. log. y³ — √})² + — hyp. log. y + √ ½ Ï + ½
+ 16 circ. arc. rad. √, tang.
2 y3
6
2 y I
And in the fourth term, 7n-1, which alfo belongs to the 7th form, cafe 2d,
and the fluent is
4
2
hyp. log. y¹² + 2 — 9 hyp. log. 1 + y*
+ ½ circ. arc. rad. √3, tang. y* — — •
But this evidently wants a correction; for when y = o, it becomes circ. arc. rad.
V3, tang.. Hence the correct fluent will be
12
L. 1+ y¹² — 8 √ ½ × L. y³ — {}² + { − L. y³ + §)² + ½
2
I
2
- 16 circ. arc. rad. ✔, tang. 2—1
2 y3
4 circ. arc. rad. 1, tang. y +
6
g L. yª
I2
+ 2/2/10 L
9 • 1 + y² + 18 circ. arc. rad. ✔3, tang.
3 y4
2y+4
+ /·
6
In which making y = 1, produces an expreffion for the ſum of the propoſed ſeries, as
in the example.
M 2
SECTION V.
66
THE SUMMATION
SECTION V..
PRO P.. X..
54. To find the Sum of a Series, the Numerators of which conftitute an Arithmetical?
Progreffion; and the Denominators confift of any Number of fimple Factors, the general.
Term being
a+bz
(Q)
p+qz.m+nz.r+sz.i+uz. &c..
From Art. 3 and 5. we have
? .
I+x
I
(A) = √xt!! *
xp:
9
X
*
xP:9+2
xP:q+3:
+
p + q
bx
+
p+29 P+39
± &c..
Multiply this equation by babp, and let the fluxion of the refulting expreffion
be defigned by prefixing the letter 9; then divide the whole by x, and there will i
arife
a:b-p : q
9. A b x
(B)
a+b.xa:b
P+9
±
a+26.x
p+29
abt 1
+
at 36.24:6+f2
+39
±&c..
Let this equation be multiplied by x
will be
m: n-a:b
*, and the fluent taken, and the refult-
"
I
(C) = √ B
Bxmina: b
*
X
==
a+b.x
p+q.m+n
m: n+1
n
+ &c.
It
a+26+:+2
p+29. m+212
a+3b.x
m:n+3.
+
•
p+39 · m + 3A..
Let this equation again be multiplied by "m:-1
x
(D) = fc.
Cx'
r: 5-min-1
S
a+3b.xris+3
p + 39. m + 3n.r+3s
risti
u+b.x
p+q.m+n.r+s
+
H
+ &c...
*, and the fluent will be
a + 2b. xristo
P+2q. m +2n.r+2s
+
Again
OF CONVERGING SERIES.
67
1 : 4-
Again multiply this equation by r ; s—I
become
x
น
(E) = √ D² : -
a+b.x
t: 4-7 : ´SI
Dx
=
+ &c. and fo on ad libitum.
*, and the fluent of the product will
p+q.m+n.r+s.t+u
H
1:4+2
a+26.x
p+29.m+2n.r+25.1+2
If therefore in the fingle terms of the feries, as alfo in their correfponding fums
(being properly rectified by fubftitution, &c.) we put x = 1, the expreffions arifing
will be the fums of the feries for the affumed number of factors of which the general
term is Q.—————————What is obſerved in Art. 10. for finding the fummatrices of thoſe
feries, is equally applicable here; as thefe general forms comprize other geometrical
ſeries, which we fhall have occafion to confider previous to thofe which immediately
coincide with the general form here given.
55. It has been afferted by feveral authors who have written on this ſubject, that
no feries of this order can poffibly be fummed, unlefs that in the general term, the
greatest exponent of the indefinite quantity or index z in the numerator, be at leaſt
two degrees lower than the greateſt exponent of the fame quantity z in the denomi-
nator; but it appears by our method, that feries of this order which have not this
property may be fummed with the fame facility as the others. Let the propoſed
ſeries therefore have two fimple factors in the denominators, in which the greateſt
dimenſion of z is only one degree higher than that in the numerator; the general
term being
a + bz
Pryz X m + n Z
The formula for the fum of this feries is
c = // S
I
n
SB*
m: n-a: b
J. Aba: bp:q
*; but B =
and A =
2
b
a
P
hence C =
X
min~p:9-1
q n
b
9
S
a
Reſolve this into two fluentials (Art. 11.) and make
b
9
P: 9
+
-
b
gn
I
J
17: 7
S² TF*
m
n
P
P
p:
17%
and it will be.
b x x x
C=
C =
qnπ
S
: 7 *
в т
{ x
вн
(M)
q 1 π
J
+
I FX
q A
S
bπ - hx
+
S
IF *
(by making 1 in the finite qu ntities)
1
IF X
the fum of the feries in infinitum.
56. Now as the particular formulæ for the fums of thefe fe ies cannot be obtained
from the œcumenical formula M (Art. 55.) by the method before laid down (Art.
I
34. 35
68
THE SUMMATION
1.4. 35. 48 ), we fhall therefore first confider thofe feries of which the fummations
may be obtained thereby, namely, when the denominators of the terms are involved
with a feries of powers. It must indeed be granted that fome feries of this order have
already been fummed by different authors, but this is effected in particular cafes only,
when the denon.inators of the terms are fo increafed by being multiplied into the
geometrical feries, as to come within the limits we have just mentioned. This whole
order of feries is, however, without exception, contained in this our general form
a+b
+
P + q • m + 11.ß
m+11.B
a+26
p+2q.m+2n.ß²
a+bz
p+yz.m+nz.ßz
And the fum in infinitum is expreſſed by
x
>
+
a+36
P+39⋅m+3n. B³
± &c.
(M) by f
xP:9
*
b π - b x
+
9 12 π5
S
B Fr
ληπ β+*
T
In which, when the figns of the propoſed ſeries are poſitive, if be a whole number,
6 greater than unity, and x = XX, the fum may be expreffed algebraically.
For then the formula becometh
b. x
X- T
ґ
BT
; which by expanding the expreffion, and taking the
T
Π
941π
B
fluent, produces
b. x.
n π
or by making ≈ = 1,
B
*
B-x
π
Ri-I„piqti
β
Bπ-2 xp: q+2
xP:9+π
Х
+
+ &c.
p + q
p+29
(M)
b. x T
T—I
7--2
π 3
β
В
I
X:
+
+
+ &c.
Nπ.
p + q
p+29
P+39
p+π q
the fum of the propofed feries in infinitum, as many terms being taken as there are
units in π
57. The algebraic fums of thefe feries may alſo be had when the figns change alter-
nately, if be an even whole number; for then the oecumenical formula M be-
cometh
(N)
b.x-π
q n n
S
B
E
П
x
xp: q
B+ x
the fluent of which being taken, and x put 1, there arifes
b.x.
T
(N)
N π
В
X:
P+9
π-2
β
3
I
+
&c.
p+29 P+39
p+xq
58. By a fimilar proceſs may the fummatrices of theſe feries be found. For in the
ལ
2
formula B (Art. 54) fubftitute for A the general fummatrix
I
B
9
xP:q
B-x. Bx
qx
(Art. 8.)
OF CONVERGING
69
SERIE S.
(Art. 8.) for feries with pofitive figns, and the refult is
b.x
пят
S
β x
25
T
x
B-x β
א
9
which being expanded by divifion and multiplication, as before, becomes
b.x
X
ng
7
S
вто
I-L
p: 9
*
ㅠ​-
* + Bπ-3 x 1:9 + 2 x + &c. x P :q + I
b.x-π
n q
7
В
SL
Taking the fluent, and making x = 1, we have
-Z-I 1:9+2
x + ß
B
7- -Z-2 p:qtati
x
x + &c. xP:q+~+~1
I
x.
7- -I
π- -2
B
Bπ-3
I
+
+
+ &c.
p + 2 q
p+39
p+q
-2
B
I
+
= + &c.
IH b.x
T
(M)
X
b. x
Nπ
N T
β
p+q
1—2—1
Bπ-
β
p+q.2+1
p+q.2+2
p + q.2+%
as many terms being taken as there are units in and z denotes the affumed number
of terms of the propofed feries.
Tr
59. By the fame method may the fummatrix be derived when the figns of the given
feries are alternately changeable from + to, provided be an even whole number.
For by writing the fummatrix
I
S
22
22
B
x
•xp: 9x
9
B+x. B²x
12
(Art. 8.) for A in the ex-
will be
preffion B, and proceeding as before, the fummatrix arifing
(Ń)
b.x
nq π
S
622 x
22
* b ; ¢ * * * * — * 8 *
B+x. B²x
which being expanded, and ordered as in the laft Art. as many terms of the refulting
expreffion must be taken as there are units in, which will then exhibit the fun of
as many pairs of terms or initial differences of the propofed feries, as there are units
in the index z.
EXAMPLE I..
60. Let the feries to be ſummed be
2
I
1.3
+
3
_3
I
4
+
3.5 9
5.7
+ &c..
27
Here it will be
a = b
= m = 1, p =
I, q = n = 2, B
3, and therefore
a
P
3
m
P
=
7. Hence x =6"
b
X-
7, and the feries
2
n
9
may be fummed algebraically.
Subftituting
70.
THE SUMMATION
Subftituting therefore theſe values in the formula M and M, the former gives,
the fum of the propoſed ſeries ad infinitum. And the latter
I
the
fummatrix for z terms.
4.1+22.3
EXAMPLE II.
61. Let the propofed feries be
49
I
61
I
+
+ &c.
125
0, § = 1, m = 4, n = 2,
g
25+12%
2·4+2%·5%
+
5 2.8 25 3. 10
b = 12, p =
37
1.6
I
Here a = 25,
b
25
7"
12
π = 2, and x = B
4, 12 = 2, B = 5; and therefore ×
and x = ẞ*.-T. Subftitute thefe values in the formula M, M, and
the former gives
the fum of the feries; and the latter
II
8
II
I
8
X
4.5*
5
+
1+% 2+2
I
>
the fummatrix for z terms.
EXAMPLE IIL
62. The feries propofed is
14
I
17
I
20
I
4.8 2
+
&c.
6.10 4
8.12 8
11+3%
2+2.6+22.2°
8
Becauſe a = 11, b = 3, p = q = n = 2, m = 6, ß = 2, x =
39 π = 2, we
have x = ẞ. x—, and confequently the algebraic fum may be obtained. Let theſe
values therefore be written in the formulæ Ń, Ń, and the fum in infinitum will be ;
and the fummatrix
I
I
I
I
+
X
22
4.2
2x+3
2+1
IV.
EXAMPLE
63. Let the feries to be fummed be
2
I
4
I
6
22
+
+
3.4
2
5.7 4
7.10
8
+ &c.
1+22
Here a = 0, b
b = 2, p = 1, q = 2, m = 1, n = 3, B = 2, from whence
26
(=
- ) is not a whole number, neither is x equal to B.; therefore the fum of
1732.2
the
OF CONVERGING SERIES.
71
the propoſed ſeries cannot be had algebraically. We must then have recourfe to the
formula M (Art. 56.)
b x
π
S
X
•p:9x
b.x
qn B-x
q N T
which, by fubftituting the above values, becomes
r
3
S* v* - 2 S* v*
2 - X
3
2-x
S
X
1:2+5
B-X
In the firſt member of this fluential expreffion make xy, and there arifes
Siv² = hyp. log.
2-3²
4
√2
y=I, and the reſult is
√ 2 j
2-y
2
23.
√2+; in which, as there is no correction neceſſary,
√2-7
But
make
4
hyp. log.
42
√2+1
√2-1
2.
3
In the latter member of the fluential put Xu, and it becomes 4
- 2u; and by writing a for 2, we fhall have
ย
and it becomes 4
2- &
4J
S
4
2
-
2-23 3a²
hyp. log. a-u +
hyp. log. a²+au+u²
3a²
8
+
3a3
circ. arc. rad. √3, tang. u + 2.
Make uo, and the correction arifing is
a
2
a
2
8
заг
8
3a3
hyp. log. a + arc. of the fame circle, tang.
a
if therefore this
2
conftant arch be taken from the arch of which the tang. is u +
a
>
there will remain
an arch of the fame circle correfponding to the tang. 3 au
4a+2u
Now make u = 1,
and reſtore the value of a, and the fum of the propofed feries will be truly expreffed
by
2 √2 L.
√2+1
√2— I
2:3
4
1:3
2
L. √2-1
L. 2
2:3
+ 2
1:3 + 1
3
3
2:3
- 2.4 L
9
L. 2 -
4 circ. arc. rad.
3'
w.
3
, tang.
3
4
3
4.2
N
or,
72
THE SUMMATION
√2+1
or, 8 L.
✓8
√2-I
+1
4
3
Film
3
3
3
3
√ 2 + 1 − 2 √ 16 L.2.
√ 2 L. √ 4 + √ 2 + 1 -
circ. arc. rad. 3, tang. 3.
3
√ 4
3
4√4
✓
64. We may now proceed with the feries firft propofed, viz. when they are not
involved with a geometrical feries, as
a+b
+
p+q. m + n
a+26
+
a+36
p + 2 q. m + 2 n p+39 • m + 3n
+ &c.
in which we may ſuppoſe ß = 1, and the fum will be expreffed by the cecumenical
formula M (Art. 55)
b
میز
qn
q n T
π
S
P
:
9
x
b = - b x
+
S
X
x
1FX
IFX
q n π
But with respect to the ſeries of this order with pofitive figns, it muſt be acknow-
ledged, that they cannot by this or any other method be brought to an alge-
braic fum. Nor indeed can the fum be exhibited either by logarithms, or circular
arcs, or by any finite formula whatever, becauſe the known quantities are in-
volved with an expreffion of an infinite magnitude. For the fluxionary quantities
the fluent of which may be
X
P: 9 x
I *
x
p: 9+π
х
by reduction become at laſt
I-X
I
hyp. log. 1-x, and making x = 1, according to the rule preſcribed, we manifeftly
derive an infinite expreffion; which indicates the impoffibility of the ſummation.
65. But when the figns of the propofed feries are alternately changeable from pofi-
tive to negative, the fummation may be obtained, either purely algebraically, or by
the arch of a circle, or by logarithms. The formula correfponding to theſe feries is
q n F
s
x
P: 9%
I + x
b π - b x
+
q n π
:
S
x
1+x
where as often as 2× =, and ™ is an uneven whole number, the fummation is given
algebraically. For the formula then becomes
b x
I + *
• xp: 9x
q n π
I + x
which by divifion, and taking the fluent, is changed into
b x
X
xP9+x
X :
xP:9+2
xP:9+3
+
xP:9+=
&c.
;
ηπ
and making
×
p+q
= 1, we have
p+29 p+39
p+πq
b x
I
I
I
I
I
(P)
X:
+
+ &c.
N T
p+q p+29
+39
+49
p + π q
as many terms being taken as there are units in .
π
66. And
OF CONVERGING SERIES.
73
66. And the fummatrix of theſe feries will be
b x
пят
22
S
p: q
X . I X
I+x
which being ordered as before, there arifeth
b x
I
(Q)
X:
ηπ p+q p+29 P+39
b x
I
X:
Nπ
p+q.2x+ I
I
I
I
+
&c.
p + = 9
I
I
+
&c.
p+9.2x+2
P+9.2x+3
I
>
p+q.2x+ Π
in which as many pairs of terms, or initial differences, of the given
feries will be fummed, as there are units in the index z.
EXAMPLE E I.
67. Let the feries to be fummed be
6
ΙΟ
14
+
2.8
4.12 6.16
18
+ &c.
8.20
2+4%
22.4+42
,
I
: ×
==
Here a 2, b = 4, p = 0, 9 = 2, m = 4, n = 4, x = 1, 1, and thence
Subſtitute theſe values in the formulæ P and Q (Art. 65, 66.); and the
2× = T.
fum arifing is, and the fummatrix =
Z
42+2
EXAMPLE II.
68. Required the fum of the feries
5
7
9
+
5. 12 10.15 15.18
II
+ &c.
20. 21
3+22
5%.9+3%
Here it will be a = 3, b = 2, p = 0, q = 5, m = 9, n = 3, and therefore
9
= T.
x
x =
2, ≈ = 3, 2× = . Theſe values being written in the formulæ P and Q, produce
I
10000
18
for the fum ad infinitum, and
I
I
18
X:
3
I
5.22+1
I
I
+
5.2≈+2
5.2x+3
the fummatrix for 2 z terms of the given feries.
EXAMPLE III.
69. Let the feries propoſed be
"
2
3
4
+
3.5 5.7 7.9
5
9.11
+ &c.
1+22.3+2≈
N 2
Here
74
THE SUMMATION
Here ab=p = 1, m = 3, n = q = 2; whence x = 1, r = 1; 2x = no-
Theſe values being fubftituted in the formulæ P and Q, give the fum ad infinitum =
I
?
12
and the fummatrix =
I
12.
I
4·3+4%
EXAM. P L E IV.
70. The feries propoſed is
I.
3
5
グ
​+
+ &ic.
•
I 2
4.5
7.8
IO. II
-1+2%
-2+3%.-1+3%
1, b = 2, p =
Here we find a =
2,9 = 3, m = −1, n = 3, and thence-
T = {}, × = {}; ; conf.quently the fummation cannot be effected algebraically, as is
>
a fractional number. But by having recourfe to the formula for the fum, namely,
bi
S
пят
xp: 9%
1+x
+
b π - b x
n q π
•xP:9+ π
I + x
*
we have, by fubftituting the above values of p, q, &c. the fum of the propofed feries
expreffed by
I
9
S
1+x
1:3 X X
3
x
1 + x
Now, make
y³, that the exponent of the flowing quantity may become a u
whole number, and the laſt expreffion is transformed to
I
3
S
3
2
1 −y + y²
the fluent of which is
circ. arc. rad. ½ √3, tang. y
y;;
which by correction becomes
circ. arc. rad. ✔3, tang.
3y
4-2y
Now, make y = 1, and the ſum of the propoſed ſeries is truly expreffed byv
I
3
circ. arc. rad. 3, tang. ..
71. Let the feries be
I
2
I. 2
3
+
+ &c.
2.3. 3.4 4.5
4
22
2.1+Z
π
Here we have a = p = 0, b = q = m = n = 1, therefore x = 0; 1; hence
the ſum of the propofed feries cannot be expreffed algebraically. But by fubftituting
thefe values in the formula, in the preceding example, and taking the fluent, there
arifes
x
hyp. log. 1+x ;
and making ≈≈ 1, the ſum of the feries ad infinitum is
1. — L. 2.
NOTES,
OF CONVERGING SERIE S.
75
NOTE S,
EXPLANATORY,
ON SECTION. V
+
+2
L
9
*
The feries-
+
A R. T. 54.
p+q. P+29
+ &c. multiplied by b
+:
+2
bx
p+q
+
bx
p+29
+ &c..
and taking the fluxion we have
1
a
b
01-0
1+1
ath
X
atab. x
+ &c..
p + Ÿ
p+29
which divided by gives the expreffion B.
a
b
+1
And a+b.x
t
a+zb.x
+ &c.
p+q
P+29
#7:
being multiplied by x
b
✰ there ariſes
11.
#
ath.x
n
a+b. x
n
p+q
p+29
9 becomes
+
+ &c..
the
76
THE SUMMATION
the Auent of which is
?n
| 2
a+b.x
111
+ 1 x p + q
H
771
+2
a+26.x
21
+ &c.
722
+ 2 × p +29
n
n
n
n.a+b *
•
n. a
a+26
X
m
+2
p+q. m +n
which divided by n gives the expreffion C.
Again this expreffion multiplied by x
p+29.m+2n
m
212
n
a+b.x x
p+q. m+n
a+b.x
+1
a+2b.x
x
It
+
p+29.m+2n
rts
p+q.m+n.
S
+ &c.
•I
✰ gives
+ &c. and the fluent is
+2
a+2b.a
+ &c.
+25
p+29. m + 2n.
S
which divided by s gives the expreffion D. And fo on at pleaſure.
ART. 55.
The formula in the preceding Art. correfponding to thefe feries is C =
I
n
S
Bx
m: n
-a:6
*, which, by writing therein the value of B
flux. of
9. Abx
a:b-p: q
x
9
Ј
p: 9
x
*
X bx
a: b—p: q
Þ:9
I
x
x
bx
a:b-p: q
I
9
IFX
+
9
-S³
P
a:bp: q·
X
p: 9 x
IFX
.b.
a
b
9
x
b
P
9
a
P
a:b-p: gi
+
X
)
we have
b
9
9
mi: n
b
P: 9
a
P
+
X
IFX
9
IFX
9
m: n—p: q—I
*
S
P: 7
111 : 71
x'
x
m: n—p: q—I
+
= C.
1
IFX
n
14
9
I
n
b
q n
*
a:b
IFX
S
a
b
9
x
Р
४.
11
And
OF CONVERGING SERIES.
77
And the first member of this fluential refolved into two, becomes
M
p
b
n
9
a
p
x
xP:9
x
b
a
P
q n
b
9
712
n
Р
7
IFx
qn
b
9
S
71
x
p: 9%
m
P
n
9
hence the whole expreffion is
hence it becomes (inaking
b x x x
bxx
q nπ
S
xp: q
9
x
b x
π+p:9
b
x
+
IFX
q n T
IF**
qn
but+p: q is evidently mn, as alfo
b
b π
>
qn
qn T
x = 1 in the finite quantity)
b x
P: 9 x
bπb x
9+
+
M.
q n T
17X
q n T
IFX
M
ART. 56.
By writing B for I in the formula in the preceding Art. we derive the expreffion
correſponding to the propofed general ſeries.
And, when the figns of the feries are pofitive, M is manifeftly
b x
S
: 9
q n T
B-X
b x
q n T
b π
+
q n T
6:9 + 1 x
bx
B-x
q n n
B-x
P:9
bx-bπ
P: 9
B-x
q n T
B-X
b x
but
b x − b
9 12
E
q n T
q n T
X B
b.x
Q H T
X 6", hence it becomes
p: 9
: 9 x
bx b
9
x
7
x x'
q n x
B-X
S³
9
X
by divifion becomes
BT-
Now
B-X
T
I
2
+ xß² ~² + x² Bi-3 + 17.3 @7++ &c.
p: q
and this multiplied by x is
9 x
I p:
X
* + Bπ-3 x pig+² + Sic.
z
*
the
78
THE SUMMATION
the fluent of which is
B1 x
I
9
1:1+2
3
X
x²:9+3
+
+
+ &c.
P
Р
+ I
+ 2
р
+3
9
9
box
I
B
.1'
hence
X
卫
​piiti
+ I
6*
7:9+2
x
+
+ &c.
Þ
+2
b. x
T
BT-1 x
X:
Σ
9
+
+ &c.
N π
p + q
p+29
the formula M.
In like manner is the formula Ń (Art. 57.) inveſtigated.
ART.
58.
I
The general fummatrix J
preffion (B)
flux. of I
シン
​X
X
B-x.fx
BX
, it becomes
9. Abx
a: b—p: q
*
•B*.
: 9
x~x
Xx
х
x. Bx
B-x
xbxa:bp:9
:
9 x
being written for A in the ex-
b
oja
४.
2
a:b
·b
+
B-x. B²
S
&
p: 9
x
B-x.B²
X
This value being ſubſtituted in the expreffion (C)
a
b
9
I
a:b-pig-x = B.
、
B
m:n-a:b
*, there arifes
m: n
b
"B"
*
x
Ъ
a
Р
+
r
m: n―p: 9
X
q n
B-x. B²
9
b
9
SⓇ
xx.xp: 9
B-X. BE
:
= C.
The latter member being refolved into two fluentials (Art. 11.) and the refult or-
dered as in note on Art. 55, 56. there will ariſe the expreffion M. By the fame
proceſs is the formula Ñ inveſtigated.
ART.
1
OF CONVERGING SERIES.
#9
ART. 60.
Here я= 6* × ×—˜ – ½, and π = 1, a whole number; therefore the ſum of
a BT Xx
"
"
this ſeries may be obtained by the formula M; and the fummatrix by M. The
values of a, b, &c. being ſubſtituted in the former, there ariſes
I I
IX 롤 ​I
2 X I
3
I
X
-
−1 + 2
4
and the latter gives
I
I
3
X
4
I +22+2 4
I
I
4+8%,3°
In like manner are the two following examples applied to their reſpective formula.
ART. 630
This example is one of thofe of which the fummation cannot be effected algebrai
cally. But by means of the formula M (Art. 56.) we find
S-2 S
I
3 2 X
for the fum of the feries. Which by writing y for x, and u for x³, becomes
&
+421 S 1 2 3 - 4 S 2-4 +28-23.
√2
2-น
The fluent of the first term is had by Emerfon's flux. form 6th, and the fecond by
form 8th, cafe 1.
In the correction of the fluent of the fecond term, the equation circ. arc. rad.
a
a
a
a
√3, tang. u +
arc. of the fame circ. rad.
2
2
2
2
= arc. rad. ✓ 3, tang.
3au
4α+2u
, appears thus,Let DH, FK (Fig. 3.) be 1 to BE, then by fim.
triang. ED: EH :: EB: EC:: EC - DC: EB - HB. Hence EC - DCE =
EB-HEB, and HEB EB - EC2 + DCE = BC+ DCE. Alfo ED: DH::
–
EB BC, and DH: EK :: HB : FB (BC), hence ex equo ED: FK:: EBH
(BC²+DCE): BC. And alternando BC+DCE: BC :: (ED) EC-DC : FK,
O
the
1
50
THE SUMMATION
•
the tang. of the difference of the arches, CF-CI, or FI. But BC =
+u, therefore
a
a
CD =
—, and CE =
2
2
3a²
a²
a u
3a²
+
+
:: 22:
u
4
4
2
4
3 a² u
a² + 1/2 a u
3 au
4a+2u
= FK.
ol
√ 3,
2
Art. 67, 68. are applied as before to the formulæ P, Q, in Art. 65. and the two
laſt examples are fummed by means of the oecumenical formula in Art. 56.
SECTION
OF CONVERGING SERIES.
81
SECTION VI.
PROP. XI.
72. To find the Sum of the Series, when the Denominators confift of three fimple
Factors, as
+
a+b
p+q.m+n.rts
a+3b
H
p+39 · m + 3n.r+35
a+26
p+2q.m +2 n
+ &c.
+2n.r+2s
a+b≈
p+qz.m+nz.r+sz
In Art. 54. we find the fum of theſe feries expreffed by
*; where C =
I
D =
S
9. Abx
a:b-p:
B =
and A =
2
I
9
I
SE
"B:
Bx
#:n-a:b
S
:9
IFX
x
Theſe values being ſubſtituted in the expreffion D, and the refult ordered according
to Art. 11. making
a
Р
b
9
the finite quantities, there arifes
= Yog
m
#
y P
=d, and x = 1 in
b x
D =
q N s T
πδ
+
:
x
qn sπ
ans.
b x
b
•
T
S
IFX
b x
Q N s T S
+
q n s n
b x
b
qns.d
IFX
0 2
*
ヒー
​p:
IFX
S
S=
#
X
IF*
+d-
And
82
THE SUMM
SUMMATION
And by reducing thoſe terms into one which have the fame coefficients, we have
(D)
b x
q n s T &
S
*
•xp: q
x
X
b x
qns T.♪
& π
m:n
X
1FX
-π
m:n
b
x
x
bu
+
IFX
q n s π d
S
p: 9
*
9%
17X
T
ans.d.
b π - b x
+
Q N s T
бать
S
бать
I
X
m: n
x
x
IFx
expreffing the ſum of the propoſed ſeries ad infinitum.
73. Whenever therefore the figns of the propofed feries are poſitive, and π, ♪, are
whole numbers, the algebraic fum may be obtained. For the firft member of the
fluential by divifion becomes
b x
Q N s T
لة
S
xP:
x²² 9 x + xp: q+1
* + &c. xp: q + — 1
*
the fluent of which being taken, and x put = 1, we have
b x
nsnd
I
I
I
I
X:
+
+
+ &c.
p + d q
p+q p+29 P+3·9
And after the fame manner will the other member of the fluential be found =
QS π
b π − b x
JTE
X:
I
m+n
I
I
I
+
+
+ &c.
m+2n m+3n
m+n.d.
Wherefore the algebraic fum of theſe feries is
(Ŕ)
b x
nsnd
I
I
x:
p + q
p+29
+ + &c. p +59
I
p+ d q
bπ - b x
I
I
+
X:
+
+ &c.
ST
QSπ. S-T
δ
m+n
m+2n
I
m + n.d-x
74. The fummatrix of theſe ſeries when the figns are pofitive may be derived as
follows. In the expreffion B =
matrix
I
9
9. Ab xa: bmp: 9
x
ſubſtitute for A the principal fum-
and let this value of B be written in the expreffion C
(Art. 54.), and again this value of C in the expreffion D, and it becomes
S
2
p: q
X x
b x
ngs π d
S
.1 X
x
+
I-X
b π — b x
n95
"b-x
I
X
1-x
which by taking the fluent, and making = 1, produces
(Ś)
bx
nsw d
OF CONVERGING SERI E S.
83
E
(Ś)
b x
I
I
I
I
X :
+
+
+ &c.
n sπ d
p+q p+29
2+39
P+89
b x
I
I
I
X:
+
+ &c.
NS T
d
p+q.x+1
p+q.2+2
p+q.x+8
b x = b x
I
I
I
I
+
X:
+
+
+ &c.
m + n
m + 2 n
m+3n
•
m+n.d
I
X:
+
•
m+n.z+1
7
b x - b x
95π
75. By a fimilar method may the formulæ for the fum and
fummatrix be obtained.
when the figns of the propofed feries are alternately + and provided that and
♪ be even whole numbers.
>
The formula for the fum in this caſe will be
I
I
+ &c.
m+n.z + 2
m+n.z+d
3
(K)
b x
I
I
I
I
X:
+
&c.
NsTd
p + q
4+29 p+31
p+dg.
bπ-bx
I
I
I
+
X:
+
&c.
9sπ.d-x
m + n
m+2n
m+3B
I
m+n.♪ ==
And the fummatrix
bx
770
(5) X
I
Γ
I'
r
+
p+q
8+28 R+39
&c.
p+ d q
bx.
I
I
I
X:
+ &c.
nsπd
Þ+9.2x+1
p+9.2x+2
p+9.2x+8
b r - b x
I.
I
I
I
+
X:
+.
m+n m+2n m+3n
&c.
m+n.d—z
T
b B8 b x
I
I
I
X:
+ &c.
Sπ
π
m+n. 2x+1 m+n. 2x+2
m+n.2z+d
in which formulæ as many terms must be taken as there are units in and ♪-
reſpectively.
EXAMPLE I.
76. Let the feries propoſed be
5
+
+
I 2.3
6
7
2.3.4 3.4.5
+ &c.
ཝཱ
4+2
I+≈.2+≈
7%
π
It will be a = 4, b = q = n = s = m = 1, p = 0, r = 2, and therefore x = 4,
≈ = 1, ♪ = 2; hence the fun of the propofed feries may be expreffed algebraically.
For by ſubſtituting theſe values in the formulæ Ŕ and έ, the fum of the ſeries comes
Out
84
THE SUMMATION
out 3, and the fummatrix
2
3
2
+
2
I
1+% 2+%
EXAMPLE II.
77. Required the fum of the feries
Z
א
I
2
3
+
+
+ &c.
2+2.4+2.6+z
3.5.7 4.6.8 5.7.9
Here we have a = 0, b = q = n = ♪ = 1, p = 2, m = 4, r = 6, and × = —
T
−2,
≈ = 2, ♪ = 4. Theſe values being written in the formula Ŕ, Ś, the former gives
31 the fum of the propofed feries; and the latter
240
the fummatrix.
31
240
I
1
+
4
X
I
I
3
3
+
3+2 4+2 5+2 6+2
EXAMPLE III.
78. Let the feries to be fummed be
2
I.3.5
It will be a = b
* = 2, d = 4.
3
4
1+%
2.4. 6 3.5.7
+
&c.
Z. 2+2.4+z
= 1,
= q = n = s = 1, p = 0, m = 2, r = 4, and thence ×
The feries may therefore be fummed algebraically. For fubftitu-
ting the above values in the formulæ Ŕ and S, the fum of the propoſed ſeries comes
3
out
32
; and the fummatrix
I
3
I
X:
8
+
4
22+1
I
22+2
3
22+3
+
EXAMPLE IV.
79. Let the feries propofed be
3
2x+4
I
4
7
Here a =
3.6.28
-2, b
+
+
6.8.35 9.10
•
+ &c.
42
-2+3x
3༤
32·4+22.21+7%
b =
= 3, p = 0, m = 4, n = 2, r = 21, s = 7, and there-
fore
>
OF CONVERGING SERIES.
85
fore × = −3, π = 2, ♪ = 3. Subſtitute thefe values in the formula Ŕ, Ś, and
there will ariſe
and the fummatrix =
the ſum =
13
3.4.7.9
13
I
+
3.4.7.9
2.3.7
I
I
I
8
X
+
EXAMPLE
3x+6 3x+3 3%+9
MPLE V.
+
22+6
80. Required the fum of the feries
I
3
5
+
+ &c;
-1+2x
3 · 4 · 10
•
Z. 1+2.4+2Z
+
2.6 2.3.8
I.2
It will be a = −1, b = s =
2, p = 0, q
2, p = 0, q = m = n = 1, r = 4, and thence.
× = − 1, * = 1, ♪ 2. The values being fubftituted in the formulæ K, Ś, the
-
=
fum will be, and the fummatrix
en/co
3
I
5
+
X
4
Z+1
x+2˚.
?
EXAMPLE VI.
81. Let the propoſed ſeries be
I.4.7
I
4
+
+
2.6.9 3.
Here we find a =
+ &c.
-2+3x
≈.2+2≈.5+2 ≈
7
8. II
2, b = 3, p = 0, q = 1, m = n = s = 2,
n = s = 2, r = 5; from
whence x = 2/42, * = 1, ♪ , which indicate that the feries cannot be had alge-
39
braically. Having recourfe therefore to the cecumenical formula Ú (Art. 72.) we
obtain, by fubftituting the above values,
5:2
I
5
X
X. x
1
8
+ X
15
3:2
I X X x *
Make x = y², and the fluent of the refult will be
5
2
X
5
5
درو
+
درو
3
درو
+ y — hyp. log. 1+y
16
y²
+ X
+
15
5.
3
2
+y-hyp. log. 1+y.
Now
86
THE SUMMATION
Now put y = I, and the fum of the feries will be
22
2
L. 2.
45
3
EXAMPLE VII.
82. Let the feries to be fummed be
4
•
I · 3 ·5
Here we have a = q
I
=
× = }, π = {, ♪ =.
T
there arifes
3
7
ΙΟ
+
&c.
1+32
2.1+22.3 + 2 z
2.5.7 3.7.9
m = 1, b = r = 3, p = 0, n == 2; from whence
s
Subftitute thefe values in the cecumenical formula Ď, and
3: 2
* Xx
I+ *
in which writing y² for x, we have
y j
7
+
4
3 S S - Z S 2
1+ y²
I
4
تو
6 I + y²
S
I + x
تو دو
I: 2
I+Y
This expreffion being transformed, and the finite fluents taken, the reſult is
2
Ss
2
S
5
7
I
y
y³ +
3
18
3
S
2yy
5
y+ y²
3
J
Ў
I+ y²
circ. arc. rad.
I,
And the fluent of the two latter terms is,
tang. y.
Hence the fum of the feries will be
hyp. log. 1+y
I
7
L. Ity
y³ +
5
5
y
circ. arc. rad. 1, tang. y; which when
3
18
3
3
23. I
y = 1, becomes
+
L. 2 -
arc. 75°, rad. 1.
18
3
83. It is unneceffary to purſue theſe ſeries any farther which have but one factor in
the numerator. For by the fame proceſs as the fummation of theſe with three factors
in the denominators have been inveſtigated, may the fums of others be obtained,
with any number of factors whatever in the denominators, by means of Propo-
fition 10, Art. 54.
NOTES,
OF CONVERGING SERIES.
87
NOT E S,
EXPLANATORY,
ON SECTION VI.
As the general formulæ Ó, Ŕ, Ś, K, and S, in this fection are derived exactly as
thoſe in the preceding one, by the method laid down in Art. 54. they need no farther
illuftration. We fhall therefore fhew the application of a few examples to thofe ge-
neral formulæ.
A R T. 76%
Here ♪ = 2, and d-- = 1, hence by writing the values of a, b, &c. in the for--
mula K (Art. 73.) there arifes
+|~
2
x=3,. the fum.
I
3
X
+
I
2
I
2
2
And ſubſtituting the ſame values in the formula Ś, we get
3
4
I
I
I
X
+
2
2
2+1
≈ +2
+ 3 x 1/2 =+= 2
x+2
the fummatrix for z terms..
P
=
3
2
2
I.
+
1+2. 2+2?
A R. T..
88
THE SUMMATION
ART. 77.
The values of p, q, &c. being ſubſtituted in the formula Ŕ (Art. 73.) produce
+
+1
4
2
I
I
I
I
8
X: + + + — to ♪ (4) terms,
3
+
I
6
4
5
to ♪― (2) terms,
4
X:
5
171
ΙΣ
31
+
the fum.
,
720
30
240'
And the fame values being written in the formula Ś, there arifes
I
5+2
+
6+%
I
I
X
4
3
1
I
I
I
1
I
+
+
5
6
+ + X
+
+
4
3+z
4+%
+ 1 x
I
I
I
I
+
นา
5
جا
I X
+
5+%
6+%
31
I
I
I
3
3
+ X
+
the fummatrix.
4
240
3+z 4+ z 5+2 6+2
To exemplify the fummatrix, let the fum of 100 terms of this feries be required,
Here ≈ 100, and the laſt expreffion becomes by ſubſtitution
31 746131
240 79483040
5712257
47689824
We have here another inftance of the fuperiority of this method of fummation in
reſpect to its elegance and fimplicity. For this feries, which is one of thofe originally
propoſed by D. Monmort, is thought by De Moivre (Miſcell. Analyt. p. 138.) to be
the most difficult of them all; and though he has given the fummation of feveral of
thoſe feries, yet this he has paffed over with barely pointing out the method whereby
it might be effected. We should alſo here difpatch this Art. were it not that a very
eminent mathematical writer had given the fum of this feries widely different from
the above, in his Meditationes Analytica, publiſhed fince this treatife of Mr. Lorgna's *.
We ſhall therefore first try the confiftency of our Author's method, by refolving the
given feries into two others, and finding the fums of each feparately, before we take
the liberty of examining the Doctor's process. Now the propofed feries may be re-
folved into three different pairs of feries, any of which will equally anſwer our
purpose; but we fhall take that which is the most obvious, viz.
Mr. Lorgna's Specimen de fericbus convergentibus was published in 1775, and Meditationes Analyticæ in
1776.
I
3.5
OF CONVERGING SERIES.
89
6
I
13-5
+
3.5.7
4.6
I
4.6.8 +6-8-5-7-9
6
+ &c.
I
I
I
1
+
+ &c.
5.7
+
3.5 4.6
3.5.7 4.6.8 5.7.9
and therefore the difference of the fums of theſe ſeries will be the fum of the propoſed
6
6
6
+
+
+ &c.
I
I
I
one
+
+
+ &c.
3.5.7 4.6.8
5.7.9
Τ
I
I
The general term of the firft feries
+ + &c. is evidently
3.5 4.6
2+2.4+2
I
which correfponds with that in Art. 13. of which the general term is
;
p+qz.m+nz
whence by compariſon, p = 2, q = n =
1, m = 4, and
•
π =2. Theſe values
I
I
I
fubftituted in the formula L (Art. 13-) give X +
3
7
the fum ad in-
2
4
24
finitum.
The fecond feries, being divided by 6 that the numerators of the terms may become
unity, is
I
I
3.5.7
+
+ 4.6.8
I
5.7.9
I
+ &c.
2+%.4+2.6+z
= 2.
Thefe
which coincides with the general form in Art. 32.
Hence by comparing the general terms, we have
I
p+qz.m+nz.r+sz
p = 2, 9 = n = 5 = 1, m = 4, r = 6, and
•
T = 4, W
= w
values being written in the formula ---Ő, Art. 33. we obtain
3
| +
5
I
+ (= 1980 = ³)
6
I
I
I
X:
143 +
+
P)
4.2
4
I
I
•
2 2
+ㅎ​(
I I
5
120
:e)
13
480
which multiplied
?
by 6, becomes
there arifes
13
Confequently, taking the fum of the latter feries from the former,
80
7
13
31
24
80
240
the fum of the propofed feries, the fame as before.
We ſhall now confider De Moivre's method of fummation, as given in his Mifcell..
Analyt. lib. 6. cap. 3. by which method it is prefumed the Doctor intended to have
fhewn the inveſtigation of the fum of this feries in his Meditationes Analytica. Now
P. 2.
this
90
THE SUMMATION
this method of fummation is--Firſt to affume a feries (A) fuch, that the indefi-
nite quantity p being taken fucceffively equal to the terms of the progreffion 0, 1, 2,
3, 4, &c. the reſult ſhall be the very feries propoſed to be fummed, commencing at
the p+th term.
Р
Then to affume another feries (B) the terms of which confift of
the indefinite quantity p, and known ones, and which are multiplied into the terms
of the progreffion x, x², x³, &c. beginning with that term of the feries (B) of which
the denominator drawn into that of the firſt term may contain the fame factors which
are found in the denominator of the firft term of the feries A. This feries is then
multiplied by fome binomial or trinomial expreffion, as x 1, x²-1, 1-ax+bxx,
I,
&c. fuch that may produce a ſeries fimilar to the first affumed one (A); in which
making the binomial or trinomial expreffiono, and tranfpofing the negative quan-
tities, there arifes a finite expreffion for the fum of this feries; which feries coincides
with A by multiplying or dividing by fome known quantity.
Now the general term of the propofed feries is
which may be refolved into theſe two,
I
a—
I
p+2.p+4.p+6
6
p+2.p+4 p+2.P+4•pto
And the feries correfponding to thefe general terms, when p is expounded in order by
the terms of the progreffion 1, 2, 3, 4, &c. are manifefly
I
I
I
I
+
3.5 4.6
+
+
5.7 6.8
+ &c.
I
and
p+2.p+4
6
6
6
I
+
3.5.7 4.5.8
+
+ &c.
5.7.9
p+2 · p + 4·p+6
Nothing therefore can be more obvious, than when De Moivre has fhewn that the
general term of the propofed feries may be refolved into two others as the above, that
he means the difference of the fums of the feries defined by thofe general terms, being
found by either the Bernoullian method of fubtraction, or his method of multiplica-
tion, fhall exhibit the fum of the propofed feries. But here the Doctor has ftrangely
miftaken the procefs; for the feries he has affumed for either term can by no means
be made to coincide with the feries defined by that general term, whatever value be
given to p. And the affuming of p (n) = o in the conclufion is abfolutely abfurd,
as that value does not reduce the firft affumed feries into that propofed to be fummed,
which it ought to do. Hence it is evident that the Doctor has fummed a quite
different feries from that he intended. For according to his affumption, the feries
will be (making no, fince it is fo taken in the value of the whole ſeries)
I
I
I
I
2.4
+
4.6
+
+
+ &c. and
6.8
8.10
6
6
6
2.4.6 4.6.8
+
+
+ &c.
6.8.10
and
OF CONVERGING SERIES.
-91
and the latter feries taken from the former, there remains
3
4. IO 12
+
•
4
5. 12. 14
5
I
+ 6.14.16
+ &c. the fum of which is
by formula
16
R in Art. 73. of this Treatife, agreeing with the Doctor's fummation.
We ſhall now give the true proceſs according to De Moivre's method.
The feries correfponding to the general terms
have been fhewn to be refpectively
I
6
p+2. p + 4 p + 2. p + 4 + p +6
I
+
3.5
I
4.6
I
6
6
+ + &c. and
5.7
+
6
3.5.7 4.6.8 5.7.9
+
+ &c.
In order then to adapt general ſeries to theſe terms, we muſt evidently retain the fac-
tors in the denominators of the terms of the numerical feries, that when the indefinite
quantity p is made equal o, the refult may exhibit the whole propoſed feries. They
may therefore be generally repreſented thus,
6
I
I
I
+
+
+ &c.
P+3·P+5
p+4.p+6
6
P+5·P+7
6
+
+
+ &c.
P+3·P+5·2+7 p+4.p+6.p+8 P+5·P+7 · p +9
where nothing can be more obvious, than if po, the feries will become refpec-
tively
I
I
I
6
6
6
+
3.5 4.6
+
+ &c. and
+
+
+ &c.
5.7
3.5.7 4.6.8
5.7.9
I
I
If p = 1, they become
+
+ &c.
4. 6
་
5.7
6
4.6.8
6
+
+ &c.
5.7.9
I
I
If p = 2, they are
6
5.7.9
+
6
6.8.10
+ &c. From
+
+ &c.
6.8
5.7
whence it appears, that if in the finite expreffions for the fums of thefe general feries,
we take p = o, we fhall have the value of the whole infinite feries. If p = 1,
I, there
will be given the fum of the feries infinitely continued, wanting the firft term. If
p2, the feries will be fummed wanting the two firft terms, and fo on at pleaſure.
Let the feries therefore affumed be
I
I
+
p+3
I
+ x +
+4 P+5
I
1
x² +
3
P+6
x³ + &c. = S,
ety
which being multiplied by x-1, there will arife
+
I
I
P+3
+4
I
1+3
* +
I
1++
x² +
P+5 pto
3
I
I
ぶ
​I
x3 &c.
+7
x² + &c.
P+5
and
?
Qe
THE SUMMATION
and by actually adding the fimilar terms, it becomes
I
I
2x
2.2.2
+
2x3
P+3 p +4 P+3 ·P+5. p+4.p+6 P+5·P+7
+
+
+ &c. =x-IXS.
Make x-1 = 0,. tranfpofe the negative terms, and divide the whole by 2, and the
refult is
I
I
I
I
+
+
+ &c. =
·
2.p+3
P+3•P+5 p+4.p+6
And if we take po, the equation becomes.
I
I
I
+ &c. =
24
+
+
3.5 4.6 5.7
In like manner the fecond feries may be
6
p+3 · p + 5·1+7
and the feries affumed
+
+
6
p+ 4. p + 6.p+8
+
+
+
I
2.P+4
6
+ &c..
P + 5 · p + 7 · p +9
•
x +
6
p+6.p+8
x² + &c. = S,
6
24 *
242
+
+
p+4.pto
p+4•ptop+3 ·P+5 · p +7 ' p +4.p+6.p+8
6
6
6
P+3•P+5 P+4.p+6 P+5.P+7
which multiplied by x-1 produces
6
P+3.P+5
+ &c. = x− I X S.
Make x-1= o, tranfpofe the negative terms, and divide the whole by 4, and there
arifes
6
+
P+3•p+5·P+7p+4.p+6.p+8
which equation, by making po, becomes
·6
+ &c. =
6
4•P+3·P+5
+
6
4.p+4.p+6
6.
6
6
+
3.5.7 4.6.8 5.7.9
+
+ &c. =
80
13, the value of this feries infinitely
31
=
240
7 13
as be-
24 80
fore; which, by an improper affumption of the feries, is erroneoufly brought out in
continued. Confequently the fum of the propoſed ſeries is
the before-mentioned treatiſe =
I
16'
It is here worth obferving that the fummatrix of this feries may from hence be
eafily deduced. For it appears from what has been faid, that whatever whole num-
ber is taken for p in the infinite expreffion for the fum of the general feries, the reſult
will give the ſum of the propofed infinite feries from the p+th term, which being
therefore taken from the value of the whole feries muſt give an expreffion for p initial
terms thereof.
The
OF CONVERGING SERIES.
93
The fummatrix of the firft feries will therefore be
6
4.p+3.p+5
of the ſecond
13
80
7
24
13
80
I
6
7
Ni
I
24 2.p+3 2.P+4
4.p+4⋅p+6
Confequently,
6
6
+
and
>
I
+
2.p+3 2.8+4 4•P+3•p+5 4·P+4·Þ+6°
is the fummatrix of the propofed feries for p term, which by reduction becomes
31
240
I
4
I
I
3
X:
+
3+1 4+P 5+p
3
6+1
as brought out in Art. 77.
We ſhall now for the ſake of variety, fhew the fummation of the fame feries by
Bernoulli's method of fubtraction.
I
I
Affume the feries + + + 6
3
4
5
I
+
+ &c.
7
from which fubtract the ſame ſeries, and we have
1
3
I
I
I
I
+ + + + + + &c.
4
5
/co
I
7
8
I
I
I
&c.
3
4
5
6
.I
2
2
2
-2
+
3
4
3.5 4.6 5.7
&c. = 0;
6.8
2
2
2
I
I
7
hence
+
+
3.5 4.6
swhich divided by 2 becomes.
+ &c. =
+ ==
5.7
3
4 12
I
I
1
3.5
6
3.5 4.6 5.7
Again aſſume +
+
4.6
+
+ &c. =
7
5.7
24
6
6
+
+ &c. from which ſubtract the fame feries,
and there remains
6
6
24
+
3.5 4.6
3.5.7
24
4.6.8
&c. = 0,
hence
24
3.5.7
+
24
4.6.8
6
6
78
+ &c. =
+
3.5
= ; which divided by 4
4.6 120
6
6
6
becomes
+
3.5.7
4.6.8
+
+ &c. =
5.7.9
78
13
480 80
as before.
In leaving this Art. I muſt obſerve, that it has been with much reluctance I have
been led to cenfure the writings of fo eminent a mathematician as the author of Medi-
tationes Analytica, &c. &c. but as it was an indifpenfible duty to my author, and to
fcientific
!
94
THE SUMMATION
fcientific truth, it is hoped that no one, who has difcernment enough to fee the pro-
priety of it, will impute it to either the arrogance of a pedant, or the cynical difpo-
fition of a modern mathematician.
ART. 78.
The values of a, b, &c. being ſubſtituted in the formula Ŕ (Art. 75.) there ariſes
I
I
I
I
I
X
+
1
2
3
4
ㅎ ​(0)
I
+
X
|
4
3
4
(=
48
=
3.
32
And the fame values written in the formula S, produce
"
the fum.
دنا الله
32
I
8
I
I
I
I
I
1
:
+
2≈+I
22+2
2x+3
22+4
I
I
X:
4
2x+3
22+4
-100
3
I
2
3
3
X:
+
+
4.
2z+! 2z+2 2x+3 22+4
Examples 4
and
5 are applied in the fame manner to the formulæ É, S. And the
two laft are fummed by means of the cecumenical formula in Art. 72. Theſe four-
examples, it is obfervable, are of that order of ſeries which have been thought incapa-
ble of ſummation; the factors in the denominators not being in the fame arithmetical.
progreffion.
SECTION VII.
OF CONVERGING SERIE S.
95
SECTION VII.
PRO P. XII.
84. To find the Sum of a Series, the Numerators of which confift of two fimple Factors,
and the Denominators of any Number of Factors whatever; the general Form being ex-
preſſed by
a+bz Xc+ez
p+qz.m+nz.r+sz.t+uz. &c.
From Art. 3. and 5. we have
I
A =
9
P: 9
x
X
xp:9+2 xP:9+3
==
=
IFX
p + q
+
P+29 P+39
+ &c.
Multiply this equation by bap, and let the fluxion of the product be defigned
by prefixing the letter 9; then divide the whole by x, and the equation arifing is
Abx
B =
9. A b xa: b~p :
q
a:6
a+b.x
p + q
+
at 26.*
p+29
0:6+1
-+
a+3b.x
+39
a:6+z
+ &c.
I
C =
n
S
m:n-a: b
a
•
Bx
x =
Let this equation be multiplied by x":n—a:b
will be
*, and the fluent taken, and the refult
+
m: x+2
+
m:n+3
a+26.x"
a+36.*
p+q.m+n p+29.m+212 p+39⋅m+3n
# &c.
This equation being again multiplied by ex
C
>
the fluxion taken, and divided
by, there arifes
9. Cex:
D =
c:eti
a+b.c+e.x
a+2b.c+2e. ×
11
+1
+ &c.
p+q.m+n
p+2q. m +27
е
This
96
THE SUMMATION
I
E =
S
√ D'
a+b.cte
-c.e
x =
This equation being multiplied by x', and the fluent taken, we get
a+2b.c+2e.x
y: S
+1
.cte.x
ㄓ
​5+2
-+ &c.
p+29.m+2n. r + 25
Again this equation multiplied by y : smt
t:
I
x
I
S.
a+b.c+e.x
a+2b.c+2e.x
1:4+2
x =
+
F = = ƒ
+ &c.
u
Ex
p+q.m +n.r+s
x, and the fluent taken, the reſult is
p+q.m+n.r+s.t+u p+2q.m +2n.r + 25.1 + 2 u
This equation multiplied by "—
X
I
G =
S
F x
ре
x =
a+2b.c+ze.xμ:1+2
p+29.m+2n.r+25
25.1+2 u
x, and the fluent taken, we have
a+b.cte. x
p+q.m+n.r+s.i+u.µ+"
x
+ &c.
+
and ſo on in infinitum; the law of progreffion being evident. Which expreffions being.
rectified by fubftitution as before, and put 1, both in the fingle terms of the
feries, and the expreffions for their fums, the refults will exhibit formulæ for the fum:
of the feries, with the affumed number of factors in the denominators.
85. Let the feries firſt propofed have three factors in the denominators; in the ge-
neral term of which the greateſt power of the indefinite quantity z in the numerator
is but one degree lower than the greateſt power of the fame quantity in the denomi-
Which order of feries is confidered by every writer on this ſubject, as abſo-
lutely impoffible to be fummed.
nator.
ris-min-
The formula for the ſums of thefe feries by the preceding Art.is E = ƒ D✩´´
Subftituting therefore fucceffively the values of D, C, B, A, and making
m
P
= λg
n
۸,
9
E =
+
TI
b x e x
ex
q n s π
a
b
р
r
p
= x;
, the refult is
9
S
J-I
X
bπ - U x x e. λ — π
q n s π
S
x
S
1x
x
9
b w e
+
q N s π
m: n
X
IFX
I+x
x
C
-I
Xx
6.
9
= (by refolving thofe
terms which contain two fluentials into two distinct ones, by Art. 10.)
p:9
κελ
E =
q n s x d.
+
IFX
b π - bx.e λπ
gnsπ.
& -π
ST
IFX
+
OF CONVERGING SERIE S.
97
+
be X d. d-x-x+xλ
1: S
qns d. J
8
Π
S ===
x
Now make x I without the fluen-
1
Fx
tial fign, as alſo π = x,
2λ; and by collecting the terms which have a common
coefficient, the expreffion becomes
be f
p: 9
x
29 n s
+ *
IF X
X x
In thoſe feries therefore which have their figns alternately + and -, fubftitute
Р
in the place of rs, and the formula for the fum will be
9
be
1 + ** X * x
1+x
(É)
29ns
which, when dis any uneven whole number, becomes
be
29ns
I
* + ** x³ + &c. x Xx
1−x+x
3
the fluent of which being taken, and x put = 1, there arifes
P: 9
* ;
(E)
be
2125
X:
I
Ι
I
+
I
I
+ &c.
p + d q
p + q p+29 P+39 P+49
86. The fummatrix of theſe feries may be derived as follows: In the formula B
I
fubftitute for A the principal fummatrix -
S
23
I
Xx
1+x
P: 9
9
(Art. 10.) which
value of B being fubftituted in C, and again this value of C in the expreffion D, and
laftly this value of D in the expreffion E, the fummatrix required will be
be
29ns
+ ² * x 1 + +³ X + pi? :
:
*
I + x
which being ordered as in the preceding Art, and ≈ put 1, it becomes
///
be
I
I
I
I
I
(上​)
2 11 $
X:
p + q
+
+ &c.
p+29
P+39
P++1
p + d q
be
T
I
I
X:
+ Sic.
p+q· 2z+d
2ns p+9.2x+1
87. In like manner if for ♪ and
P+9.2x+2
we put refpectively a and 2x, and in the expref-
fion arifing throw thofe terms into one which have the fame coefficient, we obtain
#1: 2
be
be
29ns
SP
+ * X X
I + x
2 qn s
+xxx $:9 .༢༩
1 + x
Hence if be taken any odd whole number whatſoever, we derive as before the fum
π
of this feries, as follows,
Q 3
be
(F)
21S
98
THE SUMMATION
be
I
I
I
I
(F)
X:
2ns
p + q
+
p+29 P+39
&c.
p+q
And the fummatrix is
(É)
be
x:
2 ns P+9
be
X:
2 n s
I
I
I
I
+
&c.
p+29 1+39
p+π q
I
I
I
+ &c.
P+q·2%+π
P+9.22+1 p+9.2x+2
in which as many pairs of terms of the propofed feries will be fummed as there are units.
in z.
EXAM P. LE I.
88. Let the feries propofed be
3
4. 31
7.41
10. 51
+
&c.
1+3%. 21+10%
3+5%.2+6x. 18+5%
8.8.23 13.14.28 18.20.33
It will be a 1, b = p = 3, c = 21, e = 10, m = 2, n = 6, r = 18, q
=
=5, and therefore
4
3
= x, λ
15
2
2
S
Wherefore as d (3) is an uneven whole number, the fum of the propoſed ſeries may
be expreſſed algebraically. For fubftituting theſe values in the formulæ É, E, the
former produces
97
2.8.9.13'
the fum;
and the latter
I
2
the fummatrix.
X
97
8.9.13
I
8+10%
+
I
I
13+102
18+10%
EXAMPLE II.
89. Let the feries to be fummed be
5.25
, .8.10
2
9.45
10. 14 3. 12. 18
Here we have a = 3, b
7.35
3+2%. 15+10%
+
&c.
2.6+2%.6+4%
= n = 2, þ •, q = 1, m = r = 6, c = 15, e = 10,
3
s = 4; and π = 3 = 2×, ♪ = ½ = ^ ;
therefore the feries may be fummed algebraically. Subftitute theſe values in the for-
mulæ F, F, and the reſult is, the fum adinfinitum; and 5
4
I
X:
+
4
2x + I
I
I the fummatrix.
"
22+2
2x+3
EXAM-
OF CONVERGING SERIES.
99
EXAMPLE IH.
90. Required the fum of the ſeries
10. 12
7.13.24
12 19
+
14.26
9. 15. 38 II. 17.52
8+2x·5+7=
&c.
5+2x. 11+22. 10+14%
Here it will be a = 8, b = q = n = 2, c = p = 5, e = 7, m = 11, r = 10,
3 = 2×, d= 4; hence the feries may be ſummed
s = 14, and therefore ≈ = 3 = 2×, d
algebraically. Subſtitute
will be
theſe values in the formulæ F, É, and the fum arifing
85
4.7.9. II
and the fummatrix
I.
X:
4
85
7.9.11
I
I
I
+
7+4%
9+4%
11 +42
IV.
EXAMPLE
91. The feries propoſed is
1.3
1.5.7
+
2.5
4.7.9 7.9.. II
3.7
&c.
Here we have a = 0, b = c = 1, e = n
2
and thence x =
33, ^ = 7, 8 =
be fummed purely algebraically.
2.1+22
−2+3≈•3+22.5+2%
e = n = S 2, p = 2, 9 = m = 3, r = 5,
23), and # = 13; confequently the feries cannot
19
6
T
6
But by fubftituting the values of a, b, c, &c. in
the œcumenical formula E, and making x = without the fluential fign, there
arifes
13. 19 x
2 : 3 x x + x
3:2
5 2
3
5
x
+
2. 13 1+x
19.
i+x
Now in the fecond and third terms make xy', and they become by redu&ion
ΙΟ
X
j
19
S* ; - S;
yj+
j
I + y²
X
13
33 × ƒ ; ; - S³ + S + + +
تو
==
the fluent of which being taken, and y put = 1, the refult is
552 187
arc. 45°, rad. 1.
;
741 247
And in the firft term make x = y³, and it becomes
14
S
Ў
13,19 1 + jus
13. 19 Si
14
=
13:19 147
14
I+y
13+ 19√ 1-3+1
y y
;
y
the
100
THE SUMMATION
the fluent of which is
7
hyp. log. 1+y
hyp. log. 1-y+y'
13. 19
14
13.19
arc. rad. √3, tang. y.
I 4
arifes
hyp. log. 2
4.7
circ.
3. 13. 19
circ. are rad. 3, tang. 1. Confe-
ž✓
This expreffion being corrected, and y put 1, there
4.7
13.19
3. 13. 19
quently the fum of the ſeries will be expreſſed by
552 187
arc. 45°, rad. 1, +
741 247
14
13.19
L. 2
4.7
3. 13. 19
circ. arc. rad.
√3, tang. 1.
EXAMPLE
92. Let the feries propofed to be fumined be
V.
2.2
3.5
4.8
I+Z.
+
I. I
•
5
2
3.7 3.5.9
&c.
Z
1+32
·I+ 22.3+2%
It will be a - b
= 9
q = 1, c =
3
一
​>
π- 11, d= which indicates that the fum of
From whence x = 1, λ =
I, c = m =
I, p = 0, e = r = 3, n = S=20
the feries cannot be had algebraically. Let the above values of a, b, &c. bc there-
fore written in the cecumenical formula E, which being then properly reduced, and
x put without the fluential fign, there is produced
I
x
+
3
I+x
3
16.
x
I I
3:2
x
x
16
1 + x
'
In order then to obtain the fluent of this expreffion, let x be made in the fecond
and third terms, and they become
II j
16
y
S
I I
I y³
48
3
5
16.
S
↓ + y²
the fluent of which being taken, and y put 1, we have
ΙΙ
5
arc. 45°, rad. 1.
24
16
But the fluent of the firft term is hyp. log. 1+*, or when x = 1, hyp. log. 2.
Therefore the fum of the propoſed ſeries is
I
L. 2+
3
I I
5
arc. 45°, rad. 1.
24 16
93. It might be cafily demonftrated, if neceffary, that theſe feries having three
factors in the denominators (that is when the greateft power of the index z in the
numerator is but one degree lower than the greateft power of the fame quantity in the
denominator) cannot poffibly be funmed when the figns are pofitive; the general
expreffion for the fum in that cafe involving a quantity abfolutely infinite.
NOTES,
OF CONVERGING SERIES.
101
NOTE S,
EXPLANATOR
Y,
ON SECTION VII..
Whoever has attentively confidered the mode of inveſtigating the general formulæ
in the preceding fections, and which has been fully explained in the notes thereon,
will find no difficulty in tracing out theſe in this ſection; being determined by a fimi-
lar proceſs. We here fee with what facility the fums of thofe feries, which have been
hitherto thought incapable of any general expreffion, are obtained by their application
to the general formulæ. The fums of the three firft examples are immediately de-
rived by ſubſtitution in their reſpective theorems, as before; but the two laft not
having the neceffary requifites for the algebraic fummation, are referred to the cecu-
menical formula E in Art. 85. The former of theſe we shall therefore endeavour
to illuftrate, by exhibiting the fteps of the proceſs.
The terms compared are
z. 1+2z
ART.
91.
a+bz.c+cz
−2+3%• 3+2x·5+2x
p+qx. m+nz.r+sz
from whence are deduced the values of a, b, c, &c. as in the example; thefe being
written in the cecumenical formula E, there arifes-
19
1 X 2 X2 X /
3 × 2 × 2 × 13 × 19
X X X
X
6
9
13
2
13
X2X
Xx
S
f:
*
*
6.
+
3.
6
I + *
3 X 2 X 2 X
015
X
015
13
S
I + *
!
6
of
102
THE SUMMATION
+
19
1X2X1080
3 X 2 X 2 X
19
2
7
+
2 |
6 6
3
6
3
X
015
19
13
X
6
7
:
6
x
x
1 + x
+x
ل
14
741
14
741
I 9
४
2
- ?
9
.t
x
3
x
26
I+x
I + x
x
3
Now, by making
21
26 1+*
X
in: n
1+x
+ស
5
19
+
5
19
ر
x x
I + x
X
1: $
*+ I
y³ in the first term, it becomes
*
14
3 y² j
14
-
741
1+ y³ x y²
247
S
I + y³
39
نو
and is evidently = +
1+ y³
3
Cia
y j
1-y+y
The fluent of the firft term is hyp. log. 1+y, and that of the latter, having a trino-
mial divifor, may be eafily obtained by prob. II. p. 363. Simpſon's flux. where,
by comparing the terms, we have m = 1, r = 1, a = 1, and therefore s (= cofine
of a, rad. 1.) = v. Hence the fluent will be hyp. log.
I
1-y+y+
circ.
√3
KIN
arc. rad. 1, fine
√3.Y
√1−y+ y²
•
which by note on Art.
30,
becomes
mit
2 y
hyp. log. 1-y+y2+ circ. arc. rad. 3, fine
1 −y + y²
3 y
or tang. +
(making y = 1)
or
>
4-2y
3 y
2y-4
And therefore the fluent of the first term is
14
13. 19
2. 14
L. 2
circ. arc. rad. ≥ √3, tang. 1.
3. 13. 19
-'I,
As Emerfon's forms for trinomials appear to moft readers to be fomewhat intricate,
we ſhall again fhew the inveſtigation of the fluent of the fame expreffion thereby.
The form correſponding is the 28th, from whence λ = 2, e = 1, 0 = 1, ƒ =
8 = 1, p = 4, 5 = √3, ~ = − 1 √3 + √ y, V = + √ √3+vy; and the
transformed expreffion becomes
I
v
2
X
V
// + V 2
~ + // √ 3
i
X
√3
4
$ + v²
√3
3
i
வல்
V V
4√3
+
+
1 +0²
/ + v ²
3 V
4√ 3
4 + √ ²
I
I
ย ย
√3
·V² V
√3
2 + v²
+ V2
The fluent of the first and fecond terms is had by form 4th; which, by reduction
hyp. log. 1−y+y2. The third and
fourth
and reſtoring the values of v and V, becomes
OF
103
CONVERGING
SERIES.
fourth forms belong to form 5th; and the two laft to form 11th; the fluents of which
terms being connected by their refulting figns, the values of v and V reftored, the
terms corrected, and laftly y put = 1, the refult is circ. arc. rad. 3, tang. 4,
§ √3,
as before. But as the reduction of the fluents immediately refulting from the tables to
the above forms may not be very obvious, we fhall give the proceſs at length. In
comparing the first and fecond terms with form 4th, it will be z = v (V),
4, B = 1; hence
a = 1, ß
v i
4+0²
= 1 L. 1 + v²,
V V
SV = { L. & +V², and therefore
= S
v i
+
2 +0²
v v
4 + V :
= 1 × L. & +~ + L. & + V² = { L. 4+v² x + v
++√² = 2 × L. +~° + L. § + V²
4
(reftoring the values of v and V) { L. 1−y+y².
n = 2,
Comparing the third and fourth terms with form 5th, it will be « = 2, a = 1,
= 2 circ, arc. rad. 1, tang. 2v;
ல்
I,
B = 1, and ·.·•
S₂+
S
ல்
V
rad. 1, tang. 2 V; hence
X
*
V
4 + √ ²
=2 2 circ. arc. -
4+V₁ = 2 × diff. of the arches of which
the tangents are 2v and 2 V, to rad. 1, (by note on Art. 63. and multiplying by
3)
4√3
becauſe 2 V
-
3
2
circ. arc. rad. r, tang.
or
>
2√3
1+ 4V v
ย
√3
; which is negative
2y-I
20.
Again, the two laft terms compared with form 11th, it is a = 4, 6 = 1, n = 2,
je = 1, T=0, 1; hence (the fluent of +vv) = 2 circ. arc. rad. 1,
λ =
P:
tang. 2v, as above; y = 1, :=♪ = 3; and therefore
ย
J
9. In
=
I
(ø
= V – 1 ☀ ( being the fluent of V¹ V v
= v−V+‡ • ?~?. But
V² V
like manner we find
V: V
2 circ. arc. rad. 1, tang. 2 V), hence
1 + v²
=2 circ. arc. rad. 1, tang.
2. V
1+ 4V v
ข
>
confeq. by restoring the values of v, V,
I
the Auent of the fe terms comes out I +
I
arc.
2√3
i
and multiplying by 3
rad. 1, tang.
terms =
√3
2y-I
Hence at length we have the fluent of the four latter
R.
firs
3
104
THE SUMMATION
I
v² + 1 √ 3 × &
V² + √ √ 3 × V
Ί
√3
√3
arc.
rad. 1,
1 + 0²
+ + V ²
√3
√3
tang.
2y-I
I
I
arc. rad. I, tang.
√3
But this expreffion evidently wants a correction, for when yo, it becomes
√3; and therefore this conftant part being taken
I
from the laſt fluent, there arifes
I + I
arc. tang.
√3
√3
+
2y-I √3
Τ
arc.
tang. √3, (rad. 1.) for the correct fluent (vid. Rationale Circ. Numb. p. 162.).
√3
But, (by note on Art. 63.) + arc. tang.
23√3
tang.
2y-4
which reduced to rad.
√3 — arc. tang.
+ arc.
hence the fluent becomes +
I
√3
3 (vid. note on Art. 30) is circ. arc. rad.
2y-I
circ. arc. rad. 1, tang.
23√3
21-4
2
3, tang.
3 y
3Y
or
or
ง
2
21-4
23-4
3y
4-2y
It may be of fervice to the young reader juſt to remark here, how neceffary it is to
have a precife idea of the correction of a fluent; the former method not requiring
any, while the latter would need a correction in every term, were they to be taken
feparately. For in this cafe, where the fluent is expreffed by the arch of a given cir-
cle, determined by a tangent which is expounded by fome variable quantity affected
with known ones, it is plain that the whole fluent ought to vaniſh when that variable
quantity is fuppofed to become nothing; the arch and its tangent being cotemporaneous:
If therefore there ſhould reſult ſome conftant quantity from that fuppofition, that reſult
muſt evidently be taken from the fluent firſt found to make it correct. This I know
is nothing new, but is explained in moft books on the fubject, and in none more ex-
plicitly than thoſe I have referred to; yet, notwithſtanding, we frequently meet with
errors arifing from hence in the folutions of problems, which is the reafon of my ad-
verting to it in this place. Indeed we might have avoided the trouble of deducing
the fluent in this way; for the fluent of the original expreffion is immediately
determined by form 7th, cafe 1, to be L. +3 − § L. 1−y+y² — § arc. rad. § √3,
3'7
tang.
and that without any correction. But as it was conformable to our
4-2y
Author's proceſs, and alſo gave an opportunity of elucidating Emerſon's 28th form,
which is no where done in that treatife, it was thought it would not be unacceptable
to the reader. The other members of the fluential need no explication.
?
j
1+ y³
3
ART.
OF CONVERGING SERIES.
AR T. 93.
In the oecumenical formula E (Art. 85.) put x = 1 in the finite quantities; write
alſo for p : q, μ for m: n, and, for r: s; and let P, Q, R, denote the coefficients
to the terms reſpectively. Then will the general theorem become
µ
P
f
I
x
*
+
tes;
+ R
I-X
S
I
I
but
I-X
I
I
-2
-3 &c. which being drawn into P, and
the fluent taken, there arifes PX L. I-x-PX:
In like manner the fecond term becomes
+
- I
I
+
+ &c.
ε — 2
i
QX L. 1-x
Qx:
| "
+
M
μ-
1.
y - L.
I
I
+ &c.
+ &c.
and the third RX L. 1 - - RX.
I
Hence the fluent of the whole formula is
+
Px
P+Q+R × L. 1—x—
1-
&c.
&+
&c.
зва
which, when x = 1, according to the preſcript, produces
Rx'
y
་
P
е
R
P+Q+R X ∞∞ -
&c.
&c.
&c.
E
μ
&c.
an infinite expreffion, denoting the ſum of the ſeries when the figns are pofitive.
SECTION
R 2
106
THÉ SUMMATION
SECTION VIH.
PRO P. XII.
94. To find the Sum of a Series when the Numerators confift of two, and the Denomi-
nators of four Simple Factors in Arithmetical Progreſſion; the General Term being
(M)
a+bz .c+ex
p+qz.m+nz.r+sz.1+uz
The formula correfponding to this ſeries is
E=
b x e x
qnsad
ر
I
F =
Ex
น
*, (Art. 84.)
But by Art. 85. we have
xp
:9
bπ bx.e
+
IFX
q N s T
T
bm¢3༼「
m: n
?
IFX
•
bed. Saxλ+xλ
Y:5
+
qnss.d
IFX
buen
Wherefore, putting P =
b π — b x.e.λ —π
Susub
e:
=
gnsπ.
Jain To
= ∞, and ſubſtituting this value of E in the
R =
bed.d-x monter κλ
gnsd.d-T
t
น
above expreffion (F) we have
F = = S
P
น
S time
+
R
น
S 20-2
I
*
P: 9
IFX
x
S ==
وله
+
1+*
น
Q
S
m: n
1+x
This
OF CONVERGING
107
SERIE S.
This value of F being reduced as in Art. 12, 13. and in the quantities without the
fluential fign be made
میر
1, there ariſes
P
p:
F =
1
X
x x
นฝ
ㄣ​ˋ
9
*
+
O
น
T
-T
X
Xx
x
IFX
R
I X
+
น
ய
Xx
IFX
the general fum of the feries continued ad infinitum, of which the general term is M.
95. Now, when the figns of the propoſed feries are pofitive, and e, x, d, are any
whole numbers whatſoever, the fum may be expreffed purely algebraically. For by
divifion the firft fluential becomes
น
P
S
The fecond produces
4
QI
1+x+x²+x³ + x²+ &c. x X *
Q
I+x+x²+x³+x² + &c. *
-I
X
น
W
R
S
1+x+x²+x³+x++ &c. x
x x x.
And the third
น
The fluents of thefe expreffions being taken, and x made 1, we obtain
q
(M) P4
+
+
X:
Qn
p + q
I
I
I
I
+
+
+ &c.
p+29
p+39
p+wq
I
I
I
X :
+
+
+ &c. -
m+ n
m+2n
m + n
Rs
I
I
I
I
X
น.
+
rts r+25
+
+ &c.
1+ 3
น
W
T
m + 3 n
"
rtsow
96. The fummatrix may alſo be derived by a fimilar procefs. For fince the princi
pal fummatrix
2
x~ X x
p: 9
for feries affected with pofitive figns (Art. 10.)
9
I-X
-
is evidently the expreffion for the fum of the feries in Art. 3. multiplied by 1-x
under the fluential fign, let each member of the formula F (Art. 94.) be alſo multi-
plied by 1-x, and it will become
(F)
108
THE SUMMATION
+
น
(F)
R
Р
u w
S
P: 9
x
I
X
x
+
I-X
X
•
I-X
1-X
x
е
S
Z
W
m: m
X
I
x
x
I-X
น W T
the general fummatrix required; which
being expanded, the fluent taken, and laftly x made = 1, as before, it becomes.
(Ń) P_q² % ×:
X:
น.
I
p+q.p+q.z+I
I
p + wq. p + q.z + w
+
I
p+29 · p +9.%+2
•
+ &c..
+
+
Q n² z
X:
น
W
Ι
m+n.m+n.z+I
I
I'
+
+ &c..
m+2n.m+n.z +2
m+n.w— π
—π.m+n. z+w-π
Rs² z
X:
น. W
I-.
I
+
+ &c.
rts.rts.Z+I
r+as.r+s.x+2
I
x+s.w-d.r+s.z+w-
97. By a like method may both the fum and the fummatrix of theſe feries be invef-
tigated, when the figns are alternately + and -; provided that the quantities
∞, π, and ♪, be even whole numbers. For the expreffion for the fum then comes
out
(N)
Pq
+
21 W
Q n
1+29 P+39
I
I
I
I
+
&c.
p + q
p+wq
I
I
I
I
X:
+
&c.
m + n
mn+2n
m+ 3n
mtin.. w
+
น
R s
X:
Ꮄ
I
I
I
I
+
&c.
rts
r +25
r + 3 s
rts.w-d'
+
น.
And the fummatrix is
(Ń)
2 P q² z
X:
2 w
I
I
p+q. p + q. 2x+1 p+29. p + q2x+1
I
ptwq.p+q.2x+w
+ &c.
+
OF CONVERGING SERIES.
109
2 Q n² z
나
​X:
น W
77
I
I
+ &c.
m+n.m+n. 2x+1
m+2n.m+n.2z+2
I
m+n.w-z × m+n. 2z+w-*
+
2 Rs2 z
X:
น
I
r+s.r+s. 2z+1
1
rts.w-d x rts. 2x+∞-
I
+ scc.
r+2s.rts.2%+2
in which as many pairs of terms of the propofed feries are fummed as there are units
in 2.
EXAMPLE I
98. Let the feries propofed be
I
4
+
9
+
+ &c.
N
z²
1+z.2+%·3+≈.4+≈
2.3.4.5 3.4.5.6 4.5.6.7
Here a = c = 0, b = e = p = q =
hence x =
=
-I, à - T
n = s = u = 1, m = 2, r = 3, t = 4;
w
·I, # = 1, d= 2, 3, and therefore the fum of the feries
may be expreſſed algebraically. Subftitute theſe values in the formulæ M, ŃI, and
the ſum of the ſeries in infinitum. And the latter produces
the former gives
5
36
8 | Q
I
It
X
+
2.2+z
3·3+z
16
4.4+z
the fummatrix.
EXAMPLE II
99. Let the feries to be fummed be
1.6
2.5
6.15.21
3.8
+
2.8.18.24
4. II
3.10.21.27
&c.
1+2.2+3Z
2.4+22.12+32.18+3≈
2
Here we have a = b = q = 1, c = n = 2, es=u= 3, p = 0, m = 4, r = 12,
t = 18; hence x = 1, λ=}, π = 2, ♪ = 4,6; which indicates that the
d
feries may be fummed algebraically. Wherefore fubftituting theſe values in the for-
mulæ N, Ń, the fum of the feries is found to be
17
6.8.9
. 12
and
7
110
THE SUMMATION
and the fummatrix
Z
I
X:
4.6.9
5
10
5
8
+
+
22+5
2x+1 2.22+2 3.2x+3 3.2+-3 4.2x+4
EXAMPLE IIL
100. Required the value of the feries
3.8
3.6.12 15 6.8.15.18
•
5. 13
+
+
7.18
9.10.18.24
+ &c.
1+22.3+5%
33.4+2%.9+32.12+32
:
In this example we find a = 1, b = n = 2, c = q = s = u = 3, e = 5, p = 09~
m = 4, r = 9, t = 12, and thence, λ = 3, 7 = 2, ♪ = 3, w = 4;
d
= 4; there.
fore the ſum of the feries is given algebraically. Subſtitute theſe values in the for-
mula M, M, and the fum will be
and the fummatrix
21. I'
;
36. 144
Z
I
I
X :
48
+
9.2+1
41
119
+
18.z+2 27.x+3 36.x+4
EXAMPLE IV.
101. Required the fum of the feries
2
8
18
+
+
+ &c.
222
-1+22.1+22.6+42.15+6≈
1.3.10.21 3.5.14.27 5.7.18.33
P
p =
By comparing the terms, we have a = c = 0, b = m = 1, e = n = q = 2,
1, r = u = 6, s = 4, t = 15, and thence x = 1, λ = 1,
½, T = 1, ♪ = 2,
Theſe values being written in the formula M, M, we obtain the fum
w = 3.
in infinitum; and
I
108'
Z
I
I
5
2.3.6.8×
+
the fummatrix.
22+1
3.2x+3 2x+5
102. Let the feries be
EXAMPLE V.
I 3
•
2.4
I.2.5.14 3.3.7.16
+
+
3.5
5.4.9.18
+ &c.
2.2+2
−1+22.1+%.3+22.10+4%
By
OF
III
CONVERGING
SERIE S.
×
ㅈ
​=
By comparing the general terms we find a = o, b = m = n = e = 1, c=q=&
1, r = 3, t = 10, u = 4, from whence ½, λ = {, ♪ = 2, w = 3,
= 2, p =
and
= 3; which laft value indicates that the fum of the feries cannot be exhibited
by a pure algebraic expreffion. Having recourſe therefore to the œcumenical for-
mula ŕ (Art. 94.), we obtain, by ſubſtituting the above values in the firſt and third
members thereof, and reducing the refulting expreffions as before,
But the other member becomes
I
18
I
9
S
+ 3:2 x x x
X
I-X
تو 3 و . 23 -
S ==
and making y = 1)
preſſed by
I
17
5.6.8.9
; which, by making xy', produces
I
بر
1 + y + y² • y³ ÿ
i+y
= (by drawing out the fluent,
Hence the fum of the feries is truly ex-
31
I
L. 2.
9.30
9
53
6.8.9
IL.2.
9
EXAMPLE VI.
103. Required the fum of the ſeries
2.4
6.9
10. 14
+
I.2 2.5.6 2.3.7.8 3.4.9.10
+
+ &c.
-2+4% X-1+5%
2.1+2.3+2%.4+2%
I, e
',
T
e = 5, p = 0, q = m = n = 1,
1}, 7 = 1, ♪ = †, w = 2; which in-
*, ய
Here we find a = -2, b = t = 4, c =
r = 3, s = u = 2; whence × = 11, λ =
dicates that the ſum of the feries cannot be effected purely algebraically. Let the above
x
values therefore be written in the cecumenical formula É, and it becomes by reduction
IIO
3
x
312
x
,
I + √x
making in the finite quantities. To obtain the fluent, put y², and it
will be
3:2
x
6
18
+ 3/8 + SV
X
2
+
9
110
3
s
3
S
3:2
x
220
y³
X
I+ √x
3
4
3
220
L.2
383
3
9
making y 1, there arifes
+
2
+ S
Sity
; in which
the fum of the propofed infinite
feries.
s
104. It
-1
1
II2
SUMMATION
THE
*
t
104. It would be altogether fuperfluous to purſue theſe feries any farther, as it is
evident from the four laft fections how general formula may be raiſed for the fumma-
tion of feries with any number of fimple factors in the denominators; and that not
only for thoſe ſeries already confidered, but for others having any affigned number of
factors in the numerators. I have therefore omitted a great number of ſeries of which
the fummations may be deduced from this general method; thinking it more eligible
to leave fomething for the reader's application, than to fatiate him with repetitions of
the process in enumerating examples.
NOTES,
OF CONVERGING SERIES.
113
!
NOTE
EXPLANATORY,
ON SECTION VIII.
AR T. 98.
From the values of a, b, c, &c. we find (by Art. 94.) P = 1, Q = -4, and
2
R. Hence by fubftitution in the formula M (Art. 95) we have
I
6
I
+ X: + +
I
2 X :
+ 2 x =
2
3
-
11
2
I
I
W
(to a (3) terms)
3
4
I
+
い
​(to
W
(2) terms)
4
I
W ♪
(to - (1) terms)
alco
72
374
14
9
+
5
8
369
the fum.
12
#
The fame values being written in the formula M (Art. 96.) produce
I
I
I
+
X:
3
+
+
(to w terms)
2.x+2
3.≈+3
I
I
1
2 Z X
+
(co
W
4.3+4
T
- terms)
3·2+3
4. +4
Sa
+
S,
1
114
THE SUMMATION
+
92
I
(to w-
♪ terms) =
2
4.%+4
Z
I
א
Z
I
6
X
+
2Z X
+
6
ס|א
2.2+2
3·2+3
9%
I
22+
X
2
4•≈+4
I
1 I
16
X :
+
the fummatrix.
2
2+2 3.%+3 4.2+4
The fummation of this feries is given by Stirling in his Methodus Differentialis,
p. 25. but by a mistake in the reduction of the final expreffion for the fum, it is erro-
neouſly brought out 5
24
ART.
102.
48'
e
2=1/3
R =
3
16'
and therefore the cecumenical for-
Here we find P = 5
mula É (Art. 94.) becomes
5
576
S
I
+
1-x
Now the firſt term is plainly =
I + x + x² x
1
2
alm
3
M'N
Mich
H2
I
X
I
* X
1 S* * * * - 3 S = * * *
18√
5
576
S
x3
४
I-x
Ha
64
I
I *
43
Xx *, and ** × *-* =
I-X
Xx
~−¹ ¿ + ׳ ✰ + ׳ ; and taking the fluent, 2x +
蓋
​x = x
2
४
+
33
2
2
or (making = 1) 2 +
x
3+
2 46
which multiplied by
3
5
5
5
23
576'
gives
the value of the first term.
864'
2
And the third produces
X
5
3346
3
=
;
16 hence
23
3
11
864
160
17
is
2160
the value of the firft and third terms.
In the fecond term (transformed to) is evidently =
2
I + y + y²
درو
تو رو
=I+
which drawn into y3 becomes y3 +
j
29,
and by ex-
I+y
I + y²
panding this laft term, y4 j + y² j − y j + ÿ
j
I+y
The fluent of theſe terms
is
OF CONVERGING SERIES.
115
I
I
is
ys +
33
y² + y
hyp. log. 1+y; in which making y = 1, and
5
3
2
multiplying the refult by
there arifes
31
I
L. 2.
9.30
9
17
Hence
31
+
53
I
L.2=
L. 2, is the fum of the propofed
2160 270 9
432
9
feries ad infinitum; or '0456688 &c. the proximate value thereof.
SECTION.
116
THE SUMMATION
}
SECTION IX.
PRO P. XIV.
1
105. To inveſtigate the Sums of Series being the Reciprocals of the Powers of the Na-
tural Numbers.
From Sect. II. III. IV. we have the following general formulæ, expreffing the
fums of thoſe ſeries of which the numerators are common, and the denominators con-
fifting of any number of fimple factors at pleaſure,
I
1
S=
p: q
IFX
Ι
n q
s
xP:q+1
piqti xP:942
1:1+3
x
±
+
Н
+ &c.
p+q
p+29
1.+39
x
SAR OR
P: 1
IFX
m:n+3
m11: n~p : 9-1
x
p+39 × m+372
m:ntr
p + q x m + n
± &c.
x
X
H
*
11:n+2
p + 29 X m + 2n
+
I
મ
ny s
४
r: s-m : n-I
x
S.
m: n―p : 9-1
x
S:
piq
IF
I
n q s u
S
X
ris+2
p +29 × m + 2n xr+ 25
+ &c.
A
risti
X
p + q x m + n xr+s
t2 2l 1 : 5-1
X
S
ľ: {-1}} : N-I
X
x
:S.
p: q
111: n—p: 9-1
'
ה
t:u+z
+
p + q X m + n xr+s X t+u
1 +29 × m+2n × r+es X t + 2 4
and ſo on ad libitum; the law of continuation being manifeft.
+ &c.
+
106. Now
OF CONVERGING SERIES.
117
106. Now the general term of theſe ſeries is
(e)
I
p + qz X m+nx × r+sz X t+uz X &c.
and the general term of the feries
I
I ±
+
27
I
my
2
-Im
3"
I
|m
+ + ± &c.
4"
5"
;
may be denoted by (P) where m may be any whole number, even or odd.
Hence it appears that if in the above formulæ we make p, m, r', t, &c. = 0,
n, s, u, Sic. = 1, they will become reſpectively
SE
IF X
x
S & S //
S = S = S
+
x
2
४.
+
४
3
X
3
+1
2
+
x
4
|+
"I'm
X
20
2
SSSS
3
+ &c.
+
4
~/* w/* +
It
*
2
+ &c.
+1
४
20
3
4
x
+ &c.
and
9,
=*+
+
3
4
+
+ &c.
and fo on in infinitum. If therefore in the fingle terms of the feries we put x I,
and ſuppoſe it to be fo taken in the expreffions for the fums when they are expanded
and the fluents obtained, we fhall evidently have
S=1
I7-X
I
I
+
2
3
I
+
2
SS=
It
+
J
4
W||
+
+1
I
+ &c.
նա
I
5
+
-120
+ &c.
SSS
I
= 1±
+
IFX
3
2
ايه
3
+1
I
4
+
+ &c.
3
mlin
Confequently, &c. SSS SS
I
IF
will be the fum of the feries
of which the general term is P = Q==, provided that in the fluent of the ex-
**
་
preffion be always made = 1; in which general formula as many terms muſt be
taken as there are units in the exponent m, beginning from the firſt interior fign.
107. From the fums of theſe feries which confift of the natural numbers, the tran-
fition is eaſy to the fums of the powers of even or odd numbers. For affume
P =
118
THE SUMMATION
P = 1 +
I
+
I
24 3"
Divide this equation by 2", and there arifes
P
2 n
-
I
27
I
I
+
+ &c.
4"
+ + 1/11 + &c.
4"
он
in which there are only even numbers. Subtract this equation from the firft, and the
refult is
P
P -
=
2" - I
P =
27
2"
اين
3r
-lin
+
+
+ &c.
5" 7"
n
in which are found the powers of uneven numbers only.
In l'ke manner from the feries P may the fums of the feries be inveftigated, when
the figns are alternately pofitive and negative. For let the feries for even numbers be
multiplied by 2, and fubtracted from the feries P, and it will be
2P 27-I I
P
=
PI
2n
27-I
།-
I
I
+
27
+ &e.
3n
4"
Therefore by putting the feries in which the figns change alternately Q, P will
become
2”—I— I
Q.
108. Between the affymptotes AB, AC (Fig. 1.) right-angled at A, let there be
conftituted the equilateral hyperbola DEF, from the vertex of which E is demitted to
AC the normal EG; the latus tranfverfum being = √2, or EG (AG) = 1. Let
G be the origin of the abfciffæ GI, GC, &c. which may be generally repreſented by
x, and Cc by x; and draw CF perpendicular to the affymptote in C, and cf parallel
thereto; then will the ſpatiolum FC ef (=) be the fluxion of the area of the
に
​hyperbola, and therefore the ſpace EGCF =
+
I + *
109. Now conceive the abfciffa GC to be divided into any number of parts, and
that the feveral hyperbolic spaces GEFC, &c. are divided by their correfponding ab-
ciffe GC, &c. the refulting values of which are refpectively expounded by the per-
pendiculars CL, &c. then will the curve line drawn through the extremities of theſe
perpendiculars be the firft tranfcendant hyperbola LKM, where MG is equal to GE.
Here then we ſhall have Ccx, CL =
I
X
·S.
J
I+x
and therefore the fluxion of
and the area itſelf GMLC = S₁+x
x
the area Cc Li =
x
S
I + x
S
order therefore that the fluents of
*
1+x
x
x
and
I+ *
In
I
may coincide with thoſe in
:
Art.
1
OF
119
CONVERGING
SERIES.
Art. 105. and alfo be truly expounded by the hyperbolic fpaces,. make GI GE
1, and let x be taken fuch in the fluents of the above expreffions when expanded,
and we ſhall manifeftly have the ſpace
1
I
I
I
EGIHI -
+
+
2
3
4
5
6
+ &c.
I
and MGIK = 1
+.
2
-
I'm
I
I
+
1
+ &c.
2
110. If the abciffa GĆ be taken on the other fide the point G, it will be negative,
f f
and therefore the ſpace FEGÓ
=
S
*
་
I
and ĹMGĆ = ƒ
ƒ
And in the pofitive part it may be eafily demonftrated that as the abciffe 'GI, GC,
&c. increaſe, the ordinates IK, CL, &c. decreafe, and the hyperbola LKN con-
vex towards the afſymptote; alſo that in the pofitive part the ordinates CL, &c. of
the tranfcendant hyperbola are greater than the homologous ordinates of the common
hyperbola CF, &c. but in the negative part ĆĹ is lefs than CF. Confequently the
two branches of the curve ML, MN are infinite, continually approaching the lines
AC, AO, but never meet them. And though the ſpace BAGED of the common
hyperbola is of an infinite magnitude, yet the infinitely extended ſpace OAGMN
of the tranfcendant hyperbola is abfolutely finite, being twice the ſpace MGIK, by
Art. 107. Hence we have the fums of the following feries,
I
I
I
I
I
+
2
3
+
4 5
6
+ &c. = ſpace EGIH.
I
I
+
+
+
+
+ / + &c. = ſpace BAGED = ∞.
2
3
4
5
I
+
2
I
I +
+
-I'm #1
I
+
प
I
2
3
4
I
5
62
+ &c. ſpace MGIK.
+ + &c. = 2 ſpace MGIK – OAGMN.
+ 1/2 + 1/2 +
5
111. Again every ſpace MGCL of the firft tranfcendant hyperbola being divided
by the correfponding abciffa GC, &c., and the refalis respectively reprefented
by the perpendiculars CP, &c. the curve PEQ pafting through the extremities of
thefe perpendiculars is the fecond tranfcendant hyperbola. Let.cf be produced to P
and when Ce, CP will be =
from the conftruction, and
I
*
X I+x
the fpace EGCP =
S SS ÷
1+*
T
But GI is 1, therefore produce
=
IH
*
1
120
SUMMATION
THE
IH to the new hyperbola in R, and we ſhall have
I
1
63
+ &c. fpace EGIR.
I
3
+ + &c. =
63
4
ſpace EGIR BAGEQ.
3
I
I
I
+
الا الله
+
3
4
3
I
I +
3
+
3
2
3
I
+ +
4
-| -|
5
3
5
Hence the infinitely extended ſpace BAGEQ is of a finite magnitude.
112. In like manner any ſpace as EGCP of the ſecond hyperbola is divided by
its correſponding abciffa, and the reſult expreſſed by a normal line as CS. The
curve line paffing through the extremities of theſe perpendiculars is the third tran-
fcendant hyperbola SMT. Now produce cL to s; and becauſe CS
x
X
₁ = x
by conftruction, the fluxionary ſpace Ccs S =
+ S S S
r
+ S = S & S + + +
and the ſpace MGCS
I
x
But GI 1, therefore produce IK to the new hyperbola in V, and it will be
MGCS = S + S = ƒ + S
S =
I
2
4
1
I
I
I
+
34
+
44 5°
4
I
3* 4
5
1
I
+ &c. =
Space MGIV.
64
I
8
6+
91
ſpace MGIV,
1 + 2/1/143 + + + + + &c. =
hence the ſpace OA GMT is a finite quantity.
113. By proceeding in the fame manner we diſcover the new hyperbolic fpaces
and their corresponding feries,
as follow,
I
I
+
lis
1
ilin alin l
+
+
+ &c. = ſpace EGIR.
5
I
16
+ &c. =
+
ſpace EGIR.
15
I
6+ &c. = ſpace MGIŃ.
+
+ &c. =
32
6°
fpace MGIV.
31
I
I
I
I
+
S
2
3
I'm
5
I+
I
-12
+
5
I
1
+
2
walm w/w
film fulm
+
I +
+
+
+
2
3
4
5
ī
And ſo on in infinitum, the law of progreffion being manifeft.
114. This therefore is the general theory upon which the ſeries of the reciprocals
of the powers of the natural numbers depend; being equally applicable to any dimen-
fions whatſoever. What remains then is to exhibit the numerical values of theſe hy-
perbolic ſpaces, and confequently of their correfponding feries. Now this is moſt
elegantly effected by Cotes's formula (Harm. Menf. in Tra&t. de meth. diff. p. 33.)
for
OF
121
CONVERGING
SERIES.
for a parabolic area EGIH. For if we imagine the curve EH to be the portion of
a parabola, having the fame axes and ordinates with the hyperbola EH, the curve
lines will very nearly coincide, and therefore the parabolic area come extremely near
the propoſed hyperbolic ſpace EGIH. And by the fame Cotéfian formula may the
EGÍÁ, EGÍH, &c. be determined, having the abciffæ GI, GÍ, &c. given.
Theſe areas being divided by their reſpective abciffe, the reſults IK, ÍK, &c. will
be the ordinates to the first tranfcendant hyperbola MK (Art. 109.); and the areas
MGIK, MGÍK, &c. are determined by the fame formula. Each hyperbolic
@reas
ſpace being again divided by its reſpective abciffa, the reſults IR, ÍR, &c. are the
ordinates to the ſecond tranfcendant hyperbola (Art. 111.), the feveral areas of which
are again determined by the Cotefian formula. And ſo on ad libitum.
115. From hence is conſtructed the following table, which exhibits the areas, and
the correſponding ordinates, to the 12th order of tranſcendant hyperbolas.
T 3
Hyperbolas.
8.16060
Homologous Ordinates of the Hyperbolas.
Areas.
Corresponding Infinite Series.
·76923·71428-66666·62500 ·58823-55555-52631-50000
Apollonian, EH90909 83333 7692371428 66666 62500 58823 55555 52631 50000 EGIH=6931471=1—
1ft Tranfc. MK1 953109: 16187456 84118 81093 7833475804 73473 7131769314 MGIK=8224670=1
I
2
I
N
I
+
+
I
3
4
I
+
I
I
3
4
+
I
I
اين
I
+&c.
6
I
6
•+ &c.
2d.
ER 197641 95400 93359 9140989683 88018 86451 84973 83574 82247 EGIR
3d,
4th.
5th.
-1=6949106.
MV 198791 97661 96560 *95522 94529 93583 9267491804 90966 •90167| MGIV=9470327=
•
9715596636
ER 19939098804 98239 97687 97155 96636 96135 95646 95174 94709 EGIR 9722291=1
་
MV 1·99694 99394 99103 98818 98538 98264 97996 97733 97474 97222 MGIV='9855511= I
2
I
+
2
I
2
+
wal
6th.
&c. 199847 99695 99546 ·99399 99255 99113 98972 98834 98697 98562 &c,
*9926366=1
7th.
8th.
9th,
Ioth.
11th.
$666.95666. 19666.99666. |04666.|54666.|pg666. |Sg666. |06666.56666. | 1
+0666. 1666. Ez666. zƐ666. z7666. 15666. 19666. o4666. og666.06666. I
+9766. †EE66. So+66. 44766. O£$66. |Ez966. |46966. 14466. 4+g66. Ez666. | 1
Sz966. 79966. 86966. 98466. z4466. 01866. |4+g66. £gg66. Ez666. 19666. | 1
11866. 6zg66. gt866. 99866. $8866. +0666. Ez666. z+666. 19666. og666. 1
122
انچه
I
I
+
+
I
3
1
3+
I
اين
I
I
4
I
I
4
I
+
+
+
+
I
5³
I
I
63
I
+9
S
I
5°
+ &c.
+ &c. I
I
+ &c.
6
1
I
I
4
6
+
た
​I
5
I
4 5'
99
+ &c.
+ &c.
+
27
3°
I
+ &c.
67
-I
I
I
I
I
2 3°
I'm
+
4
5
ما
Ι
I
I
I
+
ان
+
영
​H.
4
I
5°
6>
6+ + &c.
I
+&c.
610
1
4
ΟΙ
+
I
1
THE SUMMATION
•9962322=1 +
*9981151=1—
*9990368=1
*9995089=
I
ΟΙ
+
2 3
I
I L
+
I
2
I
*9997928=1
12
2
+
3
3
I
I L
12
4
4
I
I
I I
12
+
+
5
5
I
ΤΙ
I
6
I
I
[ 2
19
+&c.
+ &c.
OF
123
CONVERGING
SERIES.
116. Hence we eafily determine the fums of thefe feries when the figns are pofi
tive, by Art. 107. as follows,
I
I +
I +
+
of
4
+ &c. =
8
+ + &c. = 1'6449340
2
12
} +
1
2
+
+
37
I
I +
I +
I +
I +
+
1 + 1/1/1 +
+
I +
I +
+
IN IN IN HIN - Hľa la apa apa la
+
W/ W/m W/m w/m W/m w/m colm wine woli w/m w/w w
I
12
+
+
4
12
&c.
+ &c. 1'0004859
+ &c. 1'0002320
+
I
10
+ &c. = 1'0009918
+
of
I
I
+ &c. = 1'0040766
+ &c. = 1'0020292
+
+
+
4
S
I
+
+
+ &c. 12022358
4
+
+ &c. = 1·0823231
+ &c. = 1'0370444
+
+
37
10+ 1+
+ &c. = 1'0173431
I
+ &c. = 1.0083927
+
NOTES,
124
THE SUMMATION
NO T
T
E S,
EXPLANATORY,
ON SECTION IX.
In this fection our Author gives the finishing ftroke to his work; having exhibited
in a most elegant manner, the exact fummation of thofe feries which have been hi-
therto attempted by approximation only. It may indeed be objected, that this method
is only an approximation, as the areas of the tranfcendant curves cannot be accurately
determined. But this objection is of no weight, fince the fummations of all feries
exhibited by circular arcs and logarithms, and which can therefore be expounded in
numbers only approximately, are yet accounted finite expreffions for the fums of fuch
feries; for it is obvious that the hyperbolic ſpaces, as well as circular arcs, are abfo-
lutely finite quantities, whether we can by any means expreſs them or not. We have
therefore the fuins of theſe infinite feries truly repreſented by a finite quantity; which
has been a Defideratum with the moſt eminent mathematicians of the laſt and preſent
age. Mr. James Bernoulli could not fucceed in the moſt fimple cafe-———the recipro-
cals of fquare numbers (a). This indeed his brother Mr. John Bernoulli effected (b),
and in conjunction with Mr. Euler diſcovered the fummations of the reciprocals of even
powers; but they could proceed no farther. Afterwards Mr. De Moivre (c) inveſtigated
an approximating theorem for ſeries of this order; but the convergency is fo flow that
a great many terms are requifite to come to any tolerable degree of accuracy. Dr.
Waring (d) is the laſt author of eminence that has confidered theſe ſeries, and has
(a) Tract. de Ser. infin. p. 254.
() Suppl. ad Mifcell. Analyt,
(b) Difquifit. de fum. Ser. Quadrat. Reciprocæ, Vol. IV. p. 42.
(d) Meditationes Analyticæ, p. 487, &c.
3
collected
OF CONVERGING
SERIES.
125
collected all that has been done on the fubject; but, whatever impro ements he may
have made in other parts of the mathematics, he does not feem to have thrown any
new light on this, having trod in the fame path with Bernoulli, Euler, or De
Moivre. For the feries into which he has transformed the reciprocal feries of the
natural numbers with an indefinite index is in the fame pred.cament with De
Moivre's; being equally as operofe, and the convergency nearly as flow as collecting
the terms of the feries itſelf, whatever value be affumed for the index of the power.
-So that the honour of completing this most extenfive, and hitherto imperfect
branch of the analytic art, has been referved for the fagacity and furprizing penetra-
tion of a LORGNA.
That the ordinates IK,
ally decreaſe, is obvious.
ART. 110.
CL, &c. of the tranfcendant hyperbola LMN continu-
For let a = the hyperbolic area EGIH, and fince
a = IK by conftr. IK × IH = a, and therefore IK is a mean ordinate between
ΙΗ
EG and HI. In like manner it appears that CL is a mean ordinate between EG
and FC; but FC HI by the nat. of the apollonian hyperbola, theref. CL — ¡K.
And from hence it appears that IK, CL, &c. are greater than the homologous ordi-
nates IH, CF, &c. reſpectively. Moreover ĈĹ is a mean ordinate between GE
and ĆÊ, and ĆĹ leſs than ĈÉ, but greater than GE (GM); and therefore the
ordinates CL, &c. decreaſe, and the ordinates ĆĹ, &c. increaſe, ad infinitum.
Confequently the tranfcendant curve LMN is infinite on both fides the vertex M,
continually approaching the affymptotic lines AO, AC.
Now the general expreffion for the tranfcendant area GMLC is
S = S
x
1+x
I
I
2
3
4
I
(Art. 109.) the fluent of which is (when x is expounded by 1) 1 — +
+ &c. by Art. 106. But P-
2
(Art. 107.) that is P = 2 Q; hence
2
2
I
1
} + +
2
2
+
3
+ &c. = 2 × GMLC, a finite expreffion for the infinite
4
ſpace OAGMN. In like manner are the fums of all the feries with pofitive figns.
determined.
ART.
120
THE SUMMATION
ART. 114.
The method here adopted by our Author for determining the areas of the ſeveral
tranſcendant curves, is that of the learned Mr. Cotes, in his treatiſe De Methodo Dif-
ferentiali, Prop. VII, where he thews an approximation for the areas of curves by
means of a parabolic curve being drawn through any number of given points, which
are the extremities of ordinates determined from the equation of the propoſed curve.
Agreeable hereto the abciffa GI (= 1) has eleven ordinates applied (being the high-
eft number Mr. Cotes has carried his formulæ to), and the feveral hyperbolic ſpaces
EGIH, EGÍH, &c. which are determinable from the nature of the curve, and the
given quantities GI, Gí, &c. are divided by their reſpective abciffæ GI, Gí, &c.
and the reſults expreffed by the normal lines IK, ík, &c. Through the given
points K, K, K, &c. a parabolic curve is drawn, by prop. VI. of the abovemen-
tioned treatiſe, which will therefore very nearly coincide with the firſt tranfcendant
hyperbola K M; and the area MGIK is determined by the formula for eleven ordi-
In like manner the areas GÍKM, GIKM, GIKM, &c. are reſpectively
determined by the formulæ for 10, 9, 8, &c. ordinates. Each of theſe areas being
divided by their correfponding abciffæ GI, Gí, &c. the reſults are denoted by the
normal lines IR, ÍR, &c. and through the points R, R, R, &c. a parabolic curve
nates.
"I ///
is again drawn, the feveral areas of which EGIR, EGÍR, &c. are determined as
"/
before by the formulæ for 11, 10, 9, &c. ordinates, refpectively. And fo on ad
libitum.
As it may be fome fatisfaction to the reader, we ſhall give the computation of the
firſt tranſcendant hyperbola MGIK. As the abciffa GI (≈ 1) is divided into 10
equal parts, the abciffe GI, GI, GI, &c. are respectively = 1, '9, 8, 7, &c.
therefore in the general expreffion for the area of an hyperbola between the affymp-
totes x {{ x² + x³
X
1 ** + &c. let x be fucceffively expounded by 1, 9, ·8,
3
4
*7, *6, &c. and the areas EGIH, EGÍH, EGTH, &c. come out 69314,
"/
•641853, *587784, &c. which divided by their correfponding abciffe 1, 9, 8, &c.
produce
OF CONVERGING SERIES.
127
produce ·69314, 71317, 73473, 75804, 78334, 81093, 84116, 87456,
•91161, 95310 and 1, the ordinates IK, ÍK, ÏK, &c. to the first tranfcendant
hyperbola K M. And by the formula for 11 ordinates, the area GMIK is
16067 A + 106300 B – 48525 C + 272400 D
•822467053,
598752
260550 E + 427368 F
R =
where AGM + IK = 1•69314, the fum of the firſt and laſt ordinates;
B = 1·66627 = the fum of the ſecond and laſt but one;
C=164634 = the fum of the third and laft but two;
D163260 the fum of the fourth and laft but three;
E = 1·62450 = the fum of the fifth and laſt but four;
F81093
F = 81093 = the middle ordinate; and R = GI = 1, the interval of
the baſe between the two extreme ordinates. By the fame proceſs are the other ordi-
nates and areas inveftigated, as in the table,
U
APPENDIX.
V 3
THE
SUMMATION OF SERIES.
EXHIBITING
GENERAL FORMULE
FOR
THE SUMMING OF BOTH A FINITE AND INFINITE NUMBER OF TERMS OF
ALL POSSIBLE NUMERAL OR LITERAL SERIES WHATSOEVER.
CONTAINING
A VARIETY OF NEW THEOREMS AND IMPROVEMENTS; WITH EXAMPLES
TO EACH FORMULA.
BY
H. CLARKE,
Author of the Practical Perspective. Rationale of Circulating Numbers; with ufe-
ful Remarks on various Parts of the Mathematics. And Tranflator of Lorgna's
Specimen de Seriebus Convergentibus, with explanatory Notes.
The Doctrine of Series is actually conducive to facilitating the Computation of many interesting
Questions, which frequently occur in the Tranfactions of Mankind.
DODSON.
There is not in the whole Scope of the Mathematical Sciences, a Subject of greater Variety
and Intricacy than the Bufinefs of Series.
SIMPSON.
:
L
[ 133 ]
ADVERTISEMENT.
IN
N a ſubject ſo extenfive as the Doctrine of Series, it cannot be
expected that the following pages can exhibit diftin&t forms
for all the variety of feries which may be invented. Whenever
therefore a ſeries is propoſed to be fummed which does not imme-
diately coincide with any of thefe forms, it must be reduced thereto;
either by the addition or ſubtraction of ſome given feries; by mul-
tiplying or dividing the terms thereof; or laftly, by transforming
it into two or more other feries: And this may be always effected,
unleſs the ſum of the propofed feries be abfolutely infinite; which
circumftance is known in general from the divergency of the
terms.
As it may gratify the reader's curiofity, I have fubjoined the
names of the principal Authors who have written on this fubject;
and diſtinguiſhed with an afterifm thofe whofe works have been
confulted in compiling the following theorems. Some of thefe are
found in the Acta Eruditorum of Leipfic; Journal de Sçavans;
Memoires de l'Académie Royale des Sciences; and the Philofophical
Tranfactions of London, Berlin, St. Petersburg, Bologne, and
Upfal. Other periodical publications have not been neglected, as
Mifcellanea
$
134
ADVERTISEMENT.
Mifcellanea Curiofa Mathematica, Britiſh Oracle, the Diaries, Palla-
dium, Magazines, &c. all whofe Names are here acknowledged
that are found therein to have made any improvement in this
Branch of the Mathematics.
Halley,
*
*
* Archimedes, Arabes, D'Alembert, * Barrow, Briggs, Nicholas
Bernoulli, Daniel Bernoulli, * James Bernoulli, * John Bernoulli,
Brouuker, * De Cartes, Cragg, Clairaut, March. Condorcet,
*Cotes, Coughron, Crakelt, Collins, Alex. Cuming, * Dodfon,
* Euler, * Emerfon, Fagnanus, Le Grange, Goldbach, * James
Gregory,
Harriott, Huddens, *Huygens, Hutton,
Kepler, *Keil, Landen, * Mac-Laurin, De Lagney, * Leibnits,
* Lorgna, Ifrael Lyons, Lucas de Burgo, Manfredi, * Peter Mon-
* De Moivre, Montono, Ifaac Milner, Nichole, *Newton,
Oughtred, William Paley, * Riccati, Regnald, *Saunderſon,
*James Stirling, Slufius, Simpfon, * Brook Taylor, Varignon,
*Victa, * Wallis, Waring, John Wilfon, Wildbore, Wilkin.
mort,
*
*
*
[ 135 ]
THE
SUMMATION
OF
S
E
R
I E
S.
Note. In the following Theorems, (F) denotes the Form of the propofed feries
a, b, c, &c. the initial Terms; d, d, or A, the Common Difference, or Ratio;
z,, or Z, an Indefinite Number of Terms; (S) the Summatrix, or Sum of ≈, , or
Z Initial Terms; (2) the fum of an Infinite Number of Terms; and the Fluent
of the Expreffion to which it is prefixed, wherein the indefinite Quantity is ultimately
expounded by Unity.
I.
Series of Simple Arithmetical Numbers.
1. (F) a + a+d + a + 2d + a + 3d + a + 4d + &c. a + z− 1 × d.
Z
(E) Infinite. (S) = × 2a + z−1.d.
2
2. (F) a + a−d + a−2d + a−3d + a−4d + &c. a − z−1 × d.
a
(*) Terminates in terms, if this quantity be a whole number; but in +1
d
terms, if a fraction; confeq. z is here reſtricted.
a
d
Z
(S)
|
X 20
z− 1 . d.
2
II.
Series of Simple Geometrical Numbers.
!
>
I
3. (F) a + da + d² a + d³a + d+a + &c. d²-1 a.
a
(E)
I-d
Where d is a proper fraction, and therefore the feries decreafing. If d be a whole
number, or improper fraction, the feries is increafing; and thence the fum will be
infinite.
(5) 4 =
a.
d- I
X
111.
{
136
THE SUMMATION
III.
Series of the Powers of Arithmetical Numbers.
4. (F) m+n + m+2n + m+3n +m+4n + &c. m+zn
+
+
+I
-I
Z
nrz
(S)
+ +
r+I
12
2
3.4
↑. r — I .r- 2.r-3.r-4
2 • 3 • 4 • 5 • 6.7.6
n³ z"-5
r
113 Z
2.3.4.5.6
5.7-6
7
n' z
T.T- I.r—2.r — 3.r — 4.r.
2.3.4.5.6.7.8.5.6
r—1.r−2.r−3. r — 4. r — 5. r—6. r— 7. r—8
2' • 3 • 4 • 5 • 6.7.8.9.11.12
n° z
9 &c.
m
m²
rnm'
•I
+
r+ I n
2
3.4
I. 7 — 2
n³ m
·3 &c.
2.3.4.5.6
Continued till they terminate.
(E) Terminates when r is a whole pofitive number; but runs on fine fine when r
s negative, or a fraction.
Particular Formulæ deduced froin the above general one.
Cafe 1. 1120, 1 = 1.
5. (F) 1′ + 2′ + 3″ + 4″ + 5″ + 6″ + 7″ + 8″ + &c. z″.
— I
Z
(S)
Z
+ +
r+I 2
rz
r.r-1.7-2
r.r-I.T-2.r zor
4
Z +
&c
2.3.4.5.6.7.6
3.4 2.3.4.5.6
Hence the following formula when r = 2, 3, 4, 5, &c. fucceffively.
(F)
6. 1+2+3+4+5+&c.+%=
(S)
+2
Z
7. 1+2+3+4²+&c.+z² =
3
+
א
༢
8. 1+ 2³ +3³ +4³+&c.+z³ =
4
= A
2
W
2
+
+
|
+
N/2
+
واحه
2
= B
22
C
2
+
א
4 2
3
Z
+
of +
= D
2
3
30
18 이상​이 ​이전
​+
+
N
+
524
+
=
= E
2
12
12
6
S
Z
+
2
N N
ហ
Z
3
+
ལ
11
: F
6
42
7
א
Z
720
6
7%
z²
2
+
+
= G
2
12
24
12
00
8
א
9
+
2
227 725 223
2
|
+
+
= H
3
15
9
30
&c.
&c.
Cafe
9. 1+ 2+ + 3*+4++&c.+zª
10. 1+2+3³ +4³+&c.+z5 =
11. 1+2°+3°+4° + &c.+z°
12. 1+2²+3²+4²+&c.+z²
8
13. 172³+3º+4³ + &c. +z³
&c.
||
א
5
6
א
I
OF SERIE S.
137
·
Cafe 2.
m =
= 0.
r
14. (F) nˆ + 2n + 3 + 4 n + 5n + &c. zn.
I
Z
nrz
(S)
+ +
rti .n
3.4
Z
2
r—I.r-2. n³ z
2.3.4.5.6
-3
+ &c.
Continued till it terminates; obferving to reject the laft term when r is an odd
number.
Cafe 3. m + n = a.
15. (F) a″ + a+n” +a+2n” + a+ 3n" + &c. a + z— 1. n².
(S) 1.
Affume for many whole number leſs than a, which call b; then » — a—b.
Subſtitute thefe values of m and ʼn in the General Formula, which being then conti-
nued till both the ſeries terminate, will exhibit the fum of z terms of this ſeries.
(S) 2. z+1.a′ + Ara˜˜³n + B
7.7-I
I.2
↑ — I. † 2
a
n² + C
a
n
1.2.3
r→ 1.1 −2.7-3
+ D
a²-4n+ + &c.
Where A
x²+z
2
א
Z
ሰላ
B =
>
|
+
3
2
I.2. 3.4
3
+, C = &c. as in Caſe 1.
6
This formula gives the fum of z+1 initial terms of the ſeries.
Hence the following formule when r — 2, 3, 4, 5, &c. fucceffively.
16. (F) a² +a+n² + a+2n + a + 3n + a+4n® + &c.
(S) x+1. a² + z.z+1.an +
or (S) z+1. a² + 2 Aan + Bn².
3
2.2+1 22+ I
•
n'.
I.2.3
·3
17. (F) a³ + a+n' +a+2n² + a+ 3n² + a +4n³ + &c.
(S) ≈ +1.a³ +
z.z +1.3 an ≈.≈+1.2z+1, an²
2
+
or (5) z+1.a³ + 3 Aan + 3Вan² + Cn³.
+
2
+
18. (F) a + a +n+a+2n + a + 3n+a+4n+ &c.
(S) ≈+1. a* + 4 A a³n + 6 Ban² + 4 Can³ + Dn.
a²
·S
S
19. (F) a² + a + n² + a+ 2n² + a + 3n°² +a+4n² + Sic.
+
4
(S) +1.a² + 5 Aa+n+ 10 Ba³n² + 10 Ca² n³ + 5Dan + En.
&c.
A, B, C, &c. being as in Cafe 1.
X 2
IV.
138
THE SUMMATION
IV.
Series of Figurate Numbers.
12 n+in+2 n n+1.1+2 n + 3
•
+
•
I 2
•
3
•
I.2
•
3·4
n.n+ I
20. (F) 1 + n +
+
1.2
(S) I. 2. 3.4
2.2+1.2+2.2+3
to n terms
to n terms
+ &c.
21.
22.
Hence the following ferics when n is expounded by 1, 2, 3, 4, &c. fucceffively.
23.
24.
ம்
25.
26.
Order of the Figurate Numbers.
· 1
2.
(F)
i+i+ i + I + &c.
1 + 2 + 3 + 4 + &c.
3• 1 + 3 + 6 + 10+ &c.
4. 1+4+10+20+ &c.
5.1+5+15+35+ &c.
6.1+6+21+56+ &c.
&c.
&c.
(S)
Z.
Z.Z+I
I. 2
2.2+1.2+2
I 2
·
•
3.
≈.2+1.2+2.≈+3
1. 2. 3.4
Z.z+1.2+2.≈+3.+4
I . 2
•
3.4.5
z.x+1.2+2.2+3·≈+4
•
I 2
•
3 • 4 • 5 • 6
&c.
•
V.
Series of Compound Arithmetical Numbers.
Rectangles.
27. (F) m+e.p+e+m+ 2e.p+2e +m+ze • p+3e + &c. m+ze.p+ze.
%+I
(S) x x mp +
Xm+p.e +
≈+1.2≈+I
6
e².
2
Solids.
or (S) mpz +m+p X Ae + Be².
28. (F) m+e.p+c.q+s+m+ 2e.p+20.9+2e+m+3e.p+3e.q+3e + &c.
(S) mpqz +mp+mq+pq X Ae + m+p+q x Be + Ce³.
Biquadrates.
29. (F) m+e.pte.q+e.r+e+m+2e.p+2e.q+2e.r+2e + &c.
(S) m p q r z + mpq+mpr+pqr+mqr X Ae+mp+mq+mr+pq+pr+gr X Be
+ 11+p+q+r X Ce³ + Det.
Where the law of progreffion is evident; A, B, C, &c. being the fums of the feries
lin Order III. Cafe 1.
30. (F)
OF SERIES.
139
30. (F) m+1.m+2.m+3.&c. to z terms.
(S)
m+1.m+2.m + 3. &c. to z +1 terms, — m . m + 1. m +2. &c. to ≈ + 1 terms
Z+I
31. (F) m +2.m+3.m+4.m+5. &c. to z terms.
(S.)
m+2.m+3.m+4. &c. to z + 1 terms,
≈+1
m+1.m+2. &c. to z+1 terms
I
32. (F) m+3.m+4.m+5.m+6. &c. to ≈ terms.
(S)
m÷<.m + 4. ni + 5. &c. to z + 1 terms,
m+2.111 +3. &c. to ≈+ 1 terms
Z+I
33. (F) m+1.m+2.m+3.&c.(x)+m+2.m+3.m+4.&:c.(≈)+m+3.11 +4.m+5.Sic.
(z) to Z terms.
m + ¿.1 + 2 + 1 . &c. to z + 1 terms,
m.1+1.m+2. &c. to z + 1 terms
(S)
Z+I
Hence are deduced the following feries.
Cafe 1. m = 0.
34. (F) 1.2.3. &c. (z) + 2.3.4.&c. (≈) + 3.4.5. &c. (z) to Z terms.
(S)
35. (F)
(S)
Z.Z+1.Z+2. &c. to z+1 terms
•
Z+I
Cafe 2. m = -1.
I 3.5. &c. (z) + 3.5.7. &c. (z) + 5.7.9. &c. (z) to Z terms.
2Z−1.2Z+1.22+3. &c. to z+1 terms - 1 • 3 • 5. &c. to z terms
22+2
2 m - I
36. (F)
2 77
(S)
37. (F)
(S)
271-
2 172
I
+
2m-3
2 m
2m-5
211-7
+
2 m
+
+ &c.
2 772
ལ
I
2m
X ≈.
2 11-
τ
2 m-3
21-3
2771-5
212-5
X
+
X
+
X
+ &c.
2 11
2 m
2n
2,777
2 78
ལ
I
+
2,77
2n
Z
3 m n
X ≈.
38. (F)
140
THE SUMMATION
2 777 - I
212- I
21
I
38. (F)
X
2m-3
212→ 3
X
+
X
21-3
X
2 m
22
2 t
2. 113
2 72
2 t
+ &c.
2
א
Z
(S) 1
2 t
211
3tm
+
Where the law of progreffion in theſe feries is vifible.
Z
Z
22
Z²
3
+
+
2n
31n
X Z.
3mn
4 tm n
39. (F)
(S)
22-I
2༤བ Ι
2 %
I
X
2, m
X
I
22-3
+
X
3 3
X
2 Z
2 ??? 26
2 t
Z3
12 m t
+
22-5
2 Z
メ
​5
2 m
X
5
2 t
+ &c.
40. (F) ab + a−d × 6-8 + a−2 d x b−2d + &c.
b−d
Z.Z— I
(S) zab
x b d - ad+
•
Ι 2
Z. Z- I. 22 - I
I . 2
•
3
ds.
41. (F) abc + a-d. b-d. c-s + a−2d.b−2d.c− 2A + &c.
(S) zab.c
Z.Z-I
I. 2
Z.Z-1.22- I
X abs+aco + b c d +
XadA+bdA+cdd
I. 2 3
•2
Z
. Z
Z-I
d♪A.
2 2
•
And fo on ad libitum, the progreffion being evident.
42. (F)
A + B + C + D + &c.
Where A denotes the fum of z numbers in arith. progreffion, as in Order I, B the
fum of z+numbers, C the fum of z+2 numbers, D the fum of +35 numbers
of the fame progreffion, &c.
Z Z
Z.Z-1
(S) Zza +
2 x 5 + } } + Š
d+
× Ca +
d +
2
Z.2-1.Z – 2
1.2.3
-
2
2
55 do
VI.
O F
141
SERIE S.
"
VI.
Series of Compound Geometrical Numbers.
43. (F) b-m.c—n.
•ep + b-2m.c— 2n
EP +9
+ b−3m.c−zn.ep+²? + &c.
b-zm
c-ZN. €
epto 9
•
(E)
et
I- e9
Ι
b n + c m
mn. I + e²
X b c
+
Ι
I
१)
t
ep + q z
(S) Σ
x b-mz.c—nz
b−mz . n + c―nz
I-e9
•n
I — e²
•
m
mn. I + eq
+
9
I-e
44. (F)
a—m. b—m c-m
d-m. &c. X ep
+ a−2m.b−2m.c—2m.d—2m.&c. × e£tq
X
ep+2?
&c.
+ a−3m.b−3m.c— 3m.d—3m. &c. × e
&c.
+ a-zm.b-zm.c―zm.d-zm. &c. × ett~−1 · ¶¸
小
​(E)
I
B m
Cm². 1+ el
Dm³. 1+ e²² + 2 e²
X : A ~
+
3
I
291
I
I
+
1
&c.
.9
Em². 1+¿² + 8 ¿² + è ² í
9
I
(S) Σ
X : Á -
В
172
Cm². 1+ e
Úm³
I+ť² +2 €9
+
+ &c.
I
e9
3
I
I
1-9
13
I
Where a. b. c. d. &c. = A, b. c. d. &c. + a.c.d. &c. + a.b.d.&c. = B,
c. d. &c. + b. d. &c. = C, &c.
Alfo a = a—±m, b = b-zín, &c. p =p+zq.
And Á, É, Ć, &c. have the fame relation to a, b, c, &c. as A, B, C, &c. have to
a, b, c, Sic.
45• (F)
14.2
THE SUMMATION
15. (F) I + 2″ x + 3" ² + 4″ ·³ + 5" x¹ + &c.
x² x
I+AxBx² + Ca³ + &c. Nx"
(E)
n+I Ι
I
X
Here A = 2"
nt I.
B = 3"
22 + I
I
A-
n+1."+2
I. 2
C = 4″
n+ I
I
B-
n+1.n+2
I . 2
A
n+1.n+2.n+3
1.2. 3
12+ I
n+1.7 +2
n+1.2+2 . n
D = 5″ -
C
B.
n+ 3 A -
I
I. 2
I. 2
•
3
&c.
n+1.n+2•n+3•n+4
I. 2. 3.4
Hence when ʼn is expounded by 1, 2, 3, 4, &c. fucceffively, we have
(F)
46. 1+4x+9 x² + 16 x³ + &c.
(E)
1+Ax
I + x
or
,
3
3
I-λ
I+Ax+Bx3
I+4x+x²
47. 1+8x+27x² + 64³ + &c.
3
or
1
48. 1+16x+81x²+256x³ + &c.
3
49. 1+2³ *x+3³ x² + 45 x³ + &c.
&c.
*
I X
1+11x+11x² + *³
I-X
3
1+26x+66x²+26 x²+x²
I-X
&c.
VII.
OF SERIES.
143
VII.
Series of Compound Figurate Numbers.
50. (F) 1+ nx +
n.n+I
2
x² +
n.n+i.n+2
2. 3
x³ +
n.nti.n+2.n+3
2.3·4
.4
+&c.
I
(x)
I --- X
Hence if n, which denotes the order of the figurate numbers, be expounded by
1, 2, 3, 4, &c. fucceffively, we have the fums of the following infinite feries, com-
pounded of a ſeries of figurate numbers and a geometrical progreffion.
(F)
51. I+++ x² + x³ + ++
*4 +
+ +5
+ &c.
52. 1+2x+3x² + 4x³ +5 +4 + 6 +³ + &c.
*
53. 1+3x+6x²+10x³+15ª +21x³ + &c.
5+. 1+4x+10x²+20x³ +35 x + 56 x³ + &c.
55. 1+5x+15x²+35+³ +70~* +126x³+ &c.
156.
1+6x+21x²+56x³ + 126x+252 +&c.
&c.
VIII.
(E)
I
I-
1
I
I
I
I
I
I
1
I
&c.
Series of the Reciprocals of Figurate Numbers.
57. (F) 1 + 14
I
I. 2
+
+
1.71+ I
(E)
+
•
+
I. 2
I. 2 3
2.3.4
n.n+1.n+2 n.n+1 .n+2.n+3
1. 2. 3. 4. &c. z— 1
n.n+1.n+2. n + 3. &c. n + z− 2
11- I
n-2
Y
+ &c.
Hence
144
THE SUMMATION
I
+
+
I
I
14
+ &c.
Hence if the indefinite quantity n (which here alfo denotes the order of the Figurate
Numbers) be expounded by 1, 2, 3, 4, &c. fucceffively, the following feries
arife,
58. 1. I +
(F)
I
I
I
59.2.
I +
+
2
I
60. 3.
1 +
+
3
Ι
6
I
61. 4.
I +
---
4
I
62.5.
63. 6.
I +
I +
5
J
6
+
♡ AIO - 05
+
19
+
I
15
1
21
&c.
I
+
+
10-13-1
I
I
ΙΟ
+
+
1
4
I
20
+ &c.
+ &c.
+ &c.
+ &c.
35
56
+ &c.
&c.
@ 8 8 №
min tim mit
IX.
Series of the Reciprocals of the Powers of Arithmetical Numbers.
I
64. (F)
(F)
+
n
a
I
I
I
I
I
+
+
+
+ &c.
n
n
n
n
12
and
a-3d
a-4d
a-
Z-I. d
pr-1
-I
I
NI
n
I
1.a
d
(S)
+
=
a-2d
pr
2 a
n
I
IX nd pr+3_1 × n.n+1.n+2.ď³
pr+i
+
I
2 α a
2
3 • 4 • Bar +3
βά
ds
•
&c.
2 • 3 • 4 • 5 • 6 .ya
n
"+5
p²+5 — 1 × n .n+1.n+2.n+3.n+4
Where a 6, B = 30, y = 42, ♪ = 30, &c. being the denominators of the laft
terms of the expreffions denoted by B, D, F, H, &c. refpectively, in Order III. and
P =
a
a-zd
Hence
OF SERIES.
145
Hence the following formule when ʼn = 1, 2, 3, &c. fucceffively.
I
I
1
I
T
65. (F) -/-/- +
a
+
a-d a
+
+ &:c.
2d
a
I d
L.P
P-I
Pix d
P÷ — I × d³
(S)
+
+ &c.
d
2 a
2aa
2
ават
(S)
I
66. (F) +
(S)
a
P-I
ad
I
67. (F) +
&c.
68. (F)
a
P² I
2 a' d
I
1
a
n
+
I
a-d
a-d
I
a+d
2
n
I
I
+
+
+ &c.
2
a―31
P² - I
+
2 a²
I
a-2d
P³ — 1 x d
I
a as
—
P = 1 x 2"
+ &c.
Bas
I
+
+
+ &c.
a
2 d
a-3d
P³ — I
2a3
I
P4 - 1 X 3d
Po 1 x 5 d³
+
+ &c.
2 aa+
2 B a
+
I
+
a + zd
I - Pr
2 a
+
11
2 α a".
I
+
-1
d
a + 3 a
I
+ &c.
1 - F " + ¹ xnd
I
atz− 1 .d
I p² + ³×n.n+1.n+2.d
• All
ガー ​I
n-
I.a
d
(S)
+
1 − p "+5 x n . n+1.n+2.n+ 3 • n + +
2
•
3. 4. 5. 6. 7. Q
+5
Here P =
a
2.
3.4. B. ar+3
d³
•
&c.
a + x á
Hence when n is expounded by 1, 2, &c. fucceilively, the following formula
jarife,
|69. (F) = +
T
I
I
+
a
a + t
a + zá
+a+3d+
I
+ S.c.
— (s)
70. (F)
(S)
Stc.
I
* L.
+
I P
a d
a
+
I
P
+d
I
+
24
+
a + 4d
I - F² X d 1
2α12
1
4 Bat
+
I
I
+
+
+ &c.
a+24
a+31
P²
I - P· X d
1 P. X 23
+
2 a
x 17:
Ba
+ Sic.
Y 2
- EX
67630
Stc.
When
34.5
THE SUMMATION
When a and d are each expounded by unity, the general feries (68.) and the for-
mula for the fummatrix become
71. (F)
+
1
I
-15
2
p-I
n
+
(S)
I
+
11- I
་
3
I
11
+
I
I
+ &c. —=—=1.
12
ก
Z
4
I
prti
I
X N
- pr+з
+
Xn.nti.n+2
2
24B
+ &c.
I - pr
(=)
72. (F)
I
n
+
Where P =
2 a
I
&c. S S S S
W = |~
+
72
I-P-I
(S)
+
12- IX 2
(*)
X
x
X
1- X
continued to n terms.
If a = 1, and d = 2, the refult is,
I
5'
12
+
I - P”
2
2
I
7
+
n
1
2"
+ &c. to z terms.
I
a
I
I
-
Σ (71.).
·р"+зxn.n+1.n+2
3 Թ
B
+ &c.
In like manner may a great variety of particular feries be derived, by affuming differ-
'nt values for a and d.
73. (F) 1±
(E)
2
n
+
I
3
It
I
n
+
Univerfally,
I
4 5
71
+1
I
-15%
+ &c.
&c. S & S S S S
X
X
X
to n terms; being ultimately expounded by unity.
Or
I
74. (F) 1
1
(E)
2
+
Walm
4
I
n
+
5
I
&c.
#I
2
હ
2
11- I
I
Σ (71.).
X.
OF SERIES.
147
X.
Series of Fractional Numbers of which the Numerators and Denominators confift of Factors
in Arithmetical Progreſſion, the Number of Factors being fucceffively as the Progreffion
1, 2, 3, 4, &'c.
n n.ntp
75. (F)
102
+
+
B.B+P
n.n+p.n+ap
+
B.B+p. R+2p
n.ntp.nl
· n + 2 p.n+<p
p
B.E+p. B+2p. R+3p
+ &c.
•
η n + p
n+2p. &c. n + z-
(x)
B.B+p.B+2p. &c. B + z−1.p
n
R -p - n
Particular formulæ deduced from the above general one.
Cafe 1.
Make p = 1, B = n+2, add 1 to the form, and divide the whole by n. n+1.
I
76. (F)
+
•
N nt I
()
I
+
I
I
+
+ &c.
n+3.n+4
nti.n+2 n+2.n+3
I
n
Cafe 2.
Make p = 1, ß = n+3, add 1 to the form, and divide the whole by n.n+1.n+2.
I
77. (F)
+
n.nfi.n+ 2
(E)
I
n + 1. n + 2.n+3
I
I
+
+ &c.
n+2.n+3.n++
n.nti.2
Cafe 3.
Make 1, B =
p 1, B = " + 4, add to the form, and divide the whole by
1. n + i
nti.n+2
n+2.n+3•
78. (F)
(E)
I
+
I
n.n+1.n+2.n+3 n+1.n+2.n + 3 • " + 4
I
n.n+i.n+ 2 · 3
Where the law of continuation is manifeft.
+ &c.
XI.
!
148
SUMMATION
THE SUM M
f
XI.
+
I
I
+
p+q
P+29
Series of the Reciprocals of two Simple Factors in Arithmetical Progreffion.
79. (F)
I
p+q.m + n
92
(S) X :
ηπ
I. (E) X:
n π
I
+
+
p+29.m+2n
I
p+q.p+q.z+I
I
+ &c.
p+34. m + 3n
I
p+qz.m+nz
I
I
+ &c.
p + 2q. p + q.2+2
T
p+πq.p+q.z +T
I
+ &c.
p+q
S
m
9
n
x
Xx
2. (E)
I
пять
X
IFX
m
p
7
n
9
T
Obf. 1. When the figns of the propofed feries are affirmative, and is an affirma-
tive whole number, the upper fign takes place; and 1. (E) exhibits the fum ad infi-
nitum.
་
2. If the figns be alternately and, and an affirmative even whole number,
the lower fign takes place; and 1. (E) gives the fum. In this cafe the formula (S)
gives the fum of z initial differences of the terms of the feries, or the fum of 2 x
terms; confeq. the quantity z muſt be an even number.
3. When is a negative whole number, or an affirmative fractional number, the
figns of the feries being affirmative; or an uneven whole number, the figns being al-
ternately and --, the fummatrix (S) fails, and 2. (E) exhibits the fum ad infini-
tum; the upper fign obtaining when the figns of the feries are +, and the lower when
they are and
alternately.
}
XII.
OF SERIES.
149
1
XII.
Series of the Reciprocals of two Factors, being the Compound of an Arith. and Geom.
Progreffion.
I
76. (F)
H
P+9.B
I
ptaq.B=
I
I
+
± &c.
P+39. B³
p + zq. Bz
P
I
(S)
-|
నిజ
x x x
9
9
(E)
9
X
Х
Г.
४
Р
B² X B = x
9 %
B = x
1. The quantity ß muſt be an affirmative number, not lefs than unity.
ི་
-,
2. When the figns of the propofed feries are alternately + and −, the formula
(S) gives the fum of z differences of the initial terms; the quantity ≈ must therefore
be affumed an even number. In this cafe the affirmative fign takes place in both the
formulæ.
77. (F)
I. (E)
2. (E)
(S)
XIII.
I
X:
W
P÷9
Series of the Reciprocals of three Factors in Arith. Progreffin.
I
p+q.m+n.r+s
n Sπ
I
+
I
p + 29. m + 24. r +25
I
P + 29
+ &c.
+ &c.
I
p + = 9
I
p+qz.m+nz.r+sz
I
I
I
I
X:
+
&c.
m + n
m + 2 n
m+wn
9 s w
P
ngs T
NST
9 %
•
T
I
W
ω
X:
N Z
X:
qsw.π-w
X
I x" X x
p+q· p +9.≈ +1
I
p + zq. p + q z + 2
I
m+n.m+n.z+1_m+2n.m÷n.≈+2
9
*
IFX
I
nqs
sw
I
I
I
X
·
श
+ &c.
Xx
[1]
T
p + πq.p+q.z+%
Τ
+&c..
m+wn.m+11.~+w
7
T=
S
P
9
772
∞ =
S
72
w
I. When the figns of the propofed feries are affirmative, and, are whole num-
bers, the formula 1. (2) gives the fum in infinitum; retaining the affirmative figns.
2. When the figns are + and alternately, and a must be even whole numbers,
and the negative fign ufed. And in the formula (S) write 2 for s, and it will ex-
hibit the fum of 2≈ initial terms of the given feries, or the fum of z differences.
+
3. If the figns of the feries be +, or alternately and, when, or e, or both
are affirmative fractional numbers, the formula 2. (E) gives the fum in infinitum, the
negative fign obtaining in the former cafe, and the affirmative in the latter. In both
thefe cafes the formula (S) fails.
AIV.
150
THE SUMMATION
78. (F)
J. (E)
XIV.
Series of the Reciprocals of four Factors in Arith. Progreffion.
I
=
rs.1+16
p + q. m + n.rt
I
p+qz
I
p + 29. 2s
m + 2n . r + 25. t + 2u
I
m+nz.r+sz.1+uz
+
•
8q su
П
wn qu
I
I
T
ω
•
T
I
I
I
X:
+ &c.
T
P+9
p+29
p+π q
1
I
I
X:
+
+ &c.
T
m+n
m+2n
ttôn
I
I
I
X:
r+s
H
+ &c.
r+ws
14
wod-w
2. (E)
πnq su
+
TNSU, T
|(S)
+
•
ω
I
♪ngsu.d-
wnqsu.
9%
I
X:
W.T
nz
:x=
Sqsu.-w.π-d
SA
•
W
ω
T
•
r+25
xp: 9
X
X
S
S:
X x
W
IFX
X . t
m: n
S
Ι
I
p+q•p+9.2+1 P+29.p+9.x +2
I
I
m+n.p+q.z +1 m+2n.m + n.z + 2
I
I
1
+ &c.
+&c.:
p+πq. p +9.2 + π
I
+ &c.
m+dn.m+n.z + d
I
+ &c.z
r+ws.1+502 + w
X:
+
-W
r+s.r+s.x+i
r+2s.r+s.2+2
p
r
m
=
W
2
and &=
น
9
น
S
21
n
T =
t
1. Here, w, and
I.
must be whole numbers when the figns of the propofed feries
are affirmative; and even whole numbers when the figns change alternately. In the
latter cafe write 2z for z in the formula (S) and it will exhibit the fum of 2 z terms
of the propofed feries, or z initial differences.
2. When π, w, or d, are fractions (pofitive or negative) the formula 2. (E) gives
the fum in infinitum; and the fummatrix fails.
In
3. every cafe, when the figns of the feries are affirmative, the upper figns in the
formulæ muſt be uſed; and when they are alternately and, the lower.
XV.
OF SERIES.
151
XV.
79. (F)
a+26
p+29.m+2n
Series of Fractional Numbers, of which the Numerators are Arith. Numbers, and Denomi-
nators two fimple Factors in Arith. Progreffion.
a+b
p+q.m+n
a+zb
+ &c.
p+zq.m + z n
b x
I
I
I
I. (E) X:
+ &c.
2. (E)
ηπ
b x
n qπ
p+q
P+29
p+π q
S
p: q
x
9
I+x
x
b.π x
+
nq π
S
1+x
b x
I
(S) 1. (E)
X:
ทช
p+9.22+I
I
p+9.2x+2
I
+ &c.
p+q.2x+π
m
р
a
x =
n
9
b
응​-옴​.
T
1. When is an uneven whole number, and equal 2×, the fum is exhibited by
I. (2), and the fummatrix by (S).
2. If come out a fraction, the fum of the propoſed ſeries is obtained by the for-
mula 2. (E); but the fummatrix fails.
3. In both caſes it muſt be 2x = ; otherwiſe all the formulæ fail; as alſo if the
figns of the propoſed ſeries be affirmative.
XVI.
Series of Fractions the Numerators of which are Arith. Numbers, and Denominators twe
Factors in Arith. Progreffion multiplied into the terms of a Geom. Progreffion.
a+b
80. (F)
p + q. m + n.ß
b. x 7
1. (E)
X:
N T
b x
2. (E)
It
I-L
a+26
P+29.m+2n. B²
+ Sic.
2
β
I
+
+ &c.
p+q
q n π
S
(S) 1. (E)
X
p: 9
B = x
p + q
b.x
N π
+
b
P+29
q n T
x:
X
a+zb
p+zq.m+zn.65
S
X
B==
·
I
Β΄
-2--2
I
#1
+ &c.
p+9.2+1 $+9.2+2
p+q.≈ +=
2
152
THE SUMMATION
a
р
'72
Р
x =
b
9
n
1. The quantity 6 must be greater than unity.
T
2. When the figns of the proposed feries are +, muſt be a whole number, whe
+ and
π muſt be an even whole number. In the latter cafe write 22 for in the
fummatrix, and the fum of x initial differences will be obtained.
,
Z
3. If ≈ be a fraction, or x not equal to 6. x-, the formula 2. (E) gives the fam
in infinitum, and the fummatrix fails. The value of x may be either a whole number
or fraction.
XVII.
Series of Fractions of which the Numerators are Arith. Numbers, and Denominators three
fimple Factors in Arith. Progreffion.
a+b
81. (F)
p+q.m +n.rts
a+26
p+2q. m + 2n.r +25
a+bz
+ &c.
p+qz. m + nz
•
r+sz
b x
I
I
I
X:
±
+ &c.
NS π d
p+q
p+29
p+ 89
I. (E)
b x − b x
1
I
+
X:
9ST.
8
+-
m+ n
m+ 2n
2. (E)
(S)
T
q ns π d
I. (E)
S
b π - b x
9 ST
Π
bx
X
IF X
1
+ &c.
x
b π - b x
+
QN ST. J - T
S
m+n.d-n
-π
m:n
IX
b x
I
I
I
X:
+
+ &c.
ns w♪
T
p+q.x+1
p+q.z+d
I
X:
m+n.z+i
+1
P+9.x+2
Ι
m+n.x+2
+ &c.
I
m+n. z+d-π
go
m
a
б
π-
,
d =
X -
>
b
n
9
9
S
9
1. When the figns of the feries
2. When the ſigns are + and
ers; and in (S) write 2z for z.
are +, and ♪ must be whole numbers.
alternately,
d-
and 8-7 muſt be even whole num-
3. If any of the above requifites be wanting the formula 2. (E) takes place, and
S) fails.
XVIII.
OF SERIES.
153
XVIII.
Series of Fractions the Numerators of which confift of two fimple Factors, and the Denomi-
nators of three Factors in Arith. Progreffion.
a_bz.c+ez
p+qz.m+nz.r+sz
S=
m:n
I+x
82. (F)
a+b.cte
a+26.c+2e
+ &c.
p+q.m +n.r+s
p+2q. m + 2n. r + 25
8
bxe xx
q n s π de
P: 9
x
x
b n − b x.e.a.
•
x
+
I+x
q r s π
I. (E)
r: 5
be xd, d-x—λ+xλ
x
+
q ns d. d - π
I + x
be
I
I
I
2. (E)
2ns
(S) 2. (E)
X :
+
&c.
p+q
p+29
p + d q
be
I
X
2ns
p+9.2z+I
P+7·2x+2
a
C
d=
x
>
S
9
9
e
I
I
+ &c.
p+9.2x+d
m
n
9
1. When
X,
and ♪ (= 2^) is any uneven whole number, the formula 2. (E)
gives the fum; and (S) the fummatrix for z initial differences.
a, —
2. When ♪λ, and (= 2×) is any uneven whole number, for in the for-
mulæ 2. (E), (S), write; and the fum and fummatrix will be obtained in finite
terms as before.
3. If ♪ in the former cafe, or in the latter, come out a fraction, the fum is ob-
tained by 1. (E); and the ſummatrix fails.
4. The whole order fails when the figns of the propofed feries are pofitive.
XIX.
Z 2
154
THE SUMMATION
XIX.
Series of Fractions the Numerators of which confift of two fimple Factors, and the Denomi-
nators of four Factors in Arith Progreſſion.
a+b.cte
83. (F)
+
p+q.m +n.r+s.t+u
له
p+qz
•
. X
p: 9
a
26.c+20
p+2q. m + 2n.r+25.1+ 2 u
a+bx.cte
m+nz.r+sz.1+uz
+
е
น ω
•
T
S:
W
m:n
x
17x
+ &c.
X:
X:
I. (E)
2. (E)
P
κω
+
P
q
น.
+
+
R
น..
น
X:
•
Qn
W-
R s
I
p + q
T
I X
It
I
IFX
p+29
+ &c.
}
p + w q
I
m+n
I
I
+
+ &c.
m+ 2n
m+n.w-T
I
I
+
+ &c.
น. W
r+s r+25
rts.w
Pq2z
X
Κω
Qn² z
I
(S)
+
X:
I
I
I
+
+ &c.
p + q⋅p+q.z+I
p+29.p+q.2+2
p+wq.p+q. z+w
I
J
+&c
m+n.m+n.z+1
+
น.- T
R s² z
X
r+sor+s.x+I r+25.r+5.2 + 2
m+2n.m+n.z+2
m+n.w-π.m+7.2+w-π
I
±
I
I
+ &c.=
m
C
a
7:
=
x =
λ =
р
9
n
9
e
9
b
9
S
9
Внея
b π - b x. e. λ — π
be xd.d!
Κ
P =
Q=
q N s π J
R =
,
q n s π
d.
gns d.d-w
r+sow-dir+s.z+w
ω
W
t
-A
Р
21
9
1. When w, π,
♪ are whole numbers, and the figns pofitive; alfo when w, ™, ♪ are
even whole numbers, and the figns + and alternately, the formula 2. (E) gives
the fum, and (S) the fummatrix, regard being had to the figns as before.
2. When either w, π, or ♪, is a fractional number, the fum is exhibited by 1. (E);
and the fummatrix fails.
XX.
OF SERIES.
155
XX.
Recurring Series.
84. (F) A + B + C + D + E + &c.
or, a A + B B+yC+♪D+6E ÷ &c. = o.
о
Where a, B, y, &c. are either pofitive or negative quantities; taken in fuch manner
as to make the terms deftroy each other; and therefore the whi le feri s become = 0
BX A+ X A++ 8 × A + B + C + & X √ + B+、 +!?! + &c.
a + B + x + ô + ε + &c.
(E)
XXI.
Series of the Powers of a Quantity, the Exponents of which are in drith. Progreffion, and
the Terms respectively multiplied by a Series of Quantities a, B.,, ac.
r+n
85. (F) α.x" + Bi +7x
ax
α
(E)
+27
+ dx'+37
+ &c.
Xx
27
ay - B² x x " + a. By — ad + B.äṛ
ay-B² + By- ad × x² + Bö—~ xx
XXII.
Series of the Powers of a Binomial, the Terms of which are respectively multiplied by the
Quantities a, B, 7, d, &c.
Z-I
-I
≈12
-2
x² + ♪ da ≈-3x³ + Sic.
༤༤ 2
+ Ddx³ Xa+x
3
+&c.
86. (F) aa+Bb a²¬¹ x + y ca
(S) a × a + x² + Ń b x x x + x²
Where the differences of the quantities a, B, y, d, &c. are fuppofed to be continu-
a
+ Dcx' Xa+s
ally taken; and the firſt difference of the firſt order denoted by Ú, the firſt of the fe-
cond order by Ď, &c. That is,
n
B—a (y—B, d—y', dc.) = Ú,
y—B-B—a (d—y-y-B) = -2ßta =
α
Ď,
d—y—y—B—7—6—6—a, — d−3r+36—a = D.
&c.
as
This formula gives the fum of ≈+1 initial terms of the feries; and always termi-
ates when the laft differences of the quantities a, b, y, &c. are equal.
Particular
156
THE SUMMATION
Particular ferics deduced from the above general one.
B,,
Cafe 1.
When the values of a, B, %, d, &c. are refpectively defined by the terms of the
arith. progreffion, p, p +
= 0.
q, p + 2 q, p + 39, &c. and thence Ú
ལ=I
87. (F) pa²+p + q × b a x + p + 29 × ca
-2
= 9, and ´, ´.
ő,
2
+ &c.
-I
(S) p X a+ x + q b x Xa+x
Cafe 2.
of
Where a, 6, 7, 8, &c. are expounded by the reciprocals of an arith. progreffion,
I
&c.
I
I
as
P
p+29
2
88. (F)
a
+
р
ba
p + q
-I
X
2
Ca
+
da
+
2-
+3
+ &c.
p+29
p+39
a + x
ZA
(S)
9 x
2-1XB
X
9 x
p
P+9
a+x
p+29 a + x
Where A denotes the first term, B the fecond, C the third, &c.
X
X
P+39
9 x
a+x
&c.
89. (F) +/- 123-4
I
22
27
z
237
249
+
P
p + q
p+29
+
P+39 P+49
+ &c.
T
(E)
A Q
+
2 B Q
+
+
3CQ 4 DO
p.1+z?
9
Where Q =
;
I+29
190. (f)
б
3
€ 20+
p + q p+29 p+39 p+49
and A, B, C, &c. the firft, fecond, third, &c. terms.
a = B x + y *² = d±³ + E
Ex
+
+ &c.
&c.
α
(=)
It
Ď x
+
D x²
D x³
F
+ &c.
1 + x
1 + x
I + x
1+x
//
//
a - Ax≈
D x² - Ex
2+2
(S)
+
+
+ &c.
I X
1
**
I
D x − E x ≈ + 1
1. Where A, B, C, &c. reprefent the terms next fucceeding the zth term of the
Ď,
Ď,
progreffion a, B, y, d, &c. Ú, D, as before; and the firft difference of the
E
hiſt order B – A, E = the firſt difference of the fecond order, C-2 B+A, &c.
2. This latter formula gives the ſum of z initial terms of the feries.
91. (r)
OF SERIES.
157
α
91. (F) +
(ε)
Y
a
I
+
β
2
+
7
3
+
♡
情
​+
M
દ
درو
+ &c.
Ď
y — I
+
Ď
y -
3
+
y-
+
+ &c.
A y
(S)
D-E
Ey
+
y — I
y— I
+
D-Ey
+ &c.
3
༡-
1. Here A, B, C, &c. Ú, », É, É, &c. denote the fame as in the preceding
form.
2. The latter formula exhibits the fum of z initial terms of the feries.
XXIII.
Series of the Powers of a Binomial, divided by the Terms of a Series of any Number of
Simple Factors in Arith. Progreſſion.
92. (F)
•
༤
a
Z---I
ba
x
ca
+
+
+ &c.
I 2
3
•
r
2
•
•
3 · 4 ·
3 · 4 · 5 ... 1+ 2
-x+r
a+x
ztr.**
E) or (S)
≈+1.≈+2.2+3·
Here r muſt be a whole pofitive number.
Particular feries deduced from hence.
93. (F)
(S)
94. (F)
(S)
95. (F)
a
I
N
a
२
+
2
a
22
ba
+
I. 2. 3
ba
2
x
са
2-222
+
+ &c.
3
x+1
a
@ + x
2.3
x
x+1.x
+
Ca
3. +
+ &c.
a + x
x+2
Q
≈ +2.8²+1
ba
X
+
2+1.2+2. x²
Z-
ca
+
2.3.4 3.4
+ &c.
•
5
-~+3
a+x
≈+3
x+2
a
≈+3.a
(S)
= +3 ×
a
x+I. ≈ +2.≈+3
&c. &c. The law of continuation being evident.
XXIV.
158
THE SUMMATION
XXIV.
Series of the Powers of a Binomial, the Terms of which are respectively divided by the
Terms of a Series of Figurate Numbers.
96. (F)
Q
1
22
+
Z-I
ba
X
+
r+I
2-2 2
са
+
dax-3x3
r+1.{/r+2.3r+3
+ &c.
r+1 플 ​+2
•
(E)
-x+r
at.x
I
x+1. ½ x+2 · 3 ≈ + 3 ·
str.xh
From whence may be derived a number of feries as in the preceding order, by ex-
pounding r, by 1, 2, 3, 4, &c. fucceffively.
XXV.
Series of the Powers of a Binomial, the Terms of which are respectively multiplied by the
Terms of an Arith. Progreffion.
97. (F) ma.xˆ +m+n.ß. x² + " +m+2n.y.x
(E)
r+2n
+m+3" .d.*
+37
+ &c.
Ν
X
m-rXN+ · (= R.)
1. Here N = the ſum of the ſeries a x² + Bx”+” + yx²+21 + &c. finite or in-
finite (vid. Order XXI.).
2. Becauſe N is given in finite terms, N will be had in finite terms, and conſe-
quently the value of (E) alſo.
r
98. (F) mpax +m+n. p+n. ßx"+" + m+2n.p+2n.yx²+² + &c.
(E)
x Ŕ
Ꭱ
p-rx R +
(= 5.)
x²+² + &c.
99. (F) mpqax²+m+n.p+n.q+n. Bx" +"+m+2n. p +216.9 +2n.yx'
(E)
* S
9-7XS+
ટાંક
&c. &c.
The law of continuation being viſible.
XXVI.
OF SERIES.
159
1
XXVI.
Series of the Powers of a Quantity the Exponents of which are in Arith. Progreffion, and
the Terms refpeively divided by the Terms of the fame Progreffion.
:00. (F)
(E)
x
171
m
H
m-n
m+2n
m+37
X
x
x
+
+
+ &c.
m+n
m+ 2n
m+3r
S
x
IX
n
From hence may be deduced a variety of particular feries by affuming different values.
For m and n.
XXVII.
Series of Fractions the Numerators and Denominators of which confijt of any Number of
Factors in the fame Arith. Progreſſion, the Terms being reſpectively multiplicd into the
Terms of the Progreffion 1, x", 2", 3", &c.
X
a+n.b+n. &c.
p+n.q+n.r+n. &c.
IOI.
101. (F)
a.b.&c.
p.q.r.&c.
+
M
m-n
A
સ
✰
*
X: ±R F
+
7/2
m + 1
B
771
४
(E) X: ±R
H
9
m
x" +
a+2n.b+2n. &c.
p+2n. g +27. r+2n, &c.
12
m+2n
+
+ &c. F
711+2n
p-n
m+r
= Sice
m+n
771
X: ± R
± &c.
m
1. The quantities
p-m
>
22
11
n
+
+
9-n
#
x
&c.
r-n
q-p, r-p, &c. must be uneven whole numbers;
and m, n, a, b, c, &c. any numbers even or uneven.
2. The number of factors in each numerator of the feries muſt be the fame, and
lefs than the number of factors in the denominator; which number muſt alſo be the
fame in each term.
A 2
3. A
160
THE SUMMATION
D =
3. A = a-p.b-p.c-p. &c.
B =
q-p⋅r-pos-p. &c.'
a-s.b-s. &c.
p-s.q-s. &c.'
a-g.b-q.&c.
C =
p− q • r — q. &c.
a-r. b-r. &c.
p-r.q-r. &c.
>
&c. and R the fum of the feries in Order XXV.
Form 97.
4.
When the figns of the propofed feries are affirmative, the negative fign takes
place in every term; when they change alternately in the feries, they must be to taken
in the formula; the initial fign being or according as R is or + ; and in
this cafe the laft term of every line is
+
5. If n happens to be equal to or greater than any of the quantities p, q, r, &c.
the firft term (R) of that line is only to be taken.
X
Hence may the fummation of the following feries be deduced.
Cafe 1.
Rm, gm+n, r = m+2n, &c.
a. b. c. &c.
+
m
·
mn + n m+2n. &c.
102. (F)
a+2n.b+2n. &c.
m+2n.m+3n.m+4n.&c.
R
B
x: A +
x
४
a+n.b+n. &c.
m+n.m+2n.m+3n.&c.
Xx
C
D
+
3n
12
+
X
24
a+3n.b+zn. &c.
m+3n.m+4n.m+5n.&c.
+ &c.
11
I
+
X
n
m
m + n
mx
(E)
+
B
211
0138
+
I
x"
n
2.1
+
m
m+ n
m+2n
× x3* + &c.
X
D
X
37
&c.
Cafe 2.
OF SERIES.
161
น
A =
1.2.3.&c.v
ข
and
ย-I
Cafe 2.
I
B = − v A, C = v × 2—A, D=-9x
2
V-2
X
1—2A, &c.
2
3
denotes the number of factors in the denominator of each term.
103. (F)
I
p.p+n.p+2n. &c.
I
p+2n.p+3n . &c.
It
I
× ** +
&c.
p+n.p+2n.p+3n.
271
Xx
+ &c.
(x)
It
H
.บ
IX
11
x
m+n
x
112
x
x:R~
+
1.2.3.&c.v.n".x?
I
1.2.3. &c.v
ท
m+ n
m+2n
n
V X
VII
ย-
Ι
x
X:
F V X
X
บ
•
n
2
P
บ
2-1
ข
X
2
V-2
X
3
*
P
p
37
+1
x
·2.7
+
-n
X
*
p + n p+212
· 27
&c.
H
X
p-n
+
p + n
&c.
,
1. The upper, figns are ufed when the figns of the propofed feries are affirmative.
2. When the figns of the feries are + and
บ
alternately 1+ü is affirmative
I
or negative, according as
p-n
is an even or odd number.
n
104. (F)
a.b.c. &c.
p.q.r. s.t
t. &c.
+
Cafe 3.
a+n.b+n. &c.
+
p + n . q + n. &c.
aten.b+2n. &c.
p+2n.q+211. &c.
+ &c.
1
I
I
I
Bx: +
P p + n
-+-
+ &c.
p + 2 n
9-p
I
I
I
I
сх
+
+
+ &c.
(E)
P
p + n
p+2n
I
I
I
-D
DX
+
+
+ &c.
P
p + 2n
品
​Stc.
Here the number of factors in the denominator of each term must be greater by
two, at leaft, than the number of thofe in the numerator.
A a 2
XXVIII,
162
THE SUMMATION
XXVIII.
Series of the Powers of a Quantity the Exponents of which are in Arith. Progreſſion, and
the Coefficients a Series of Fractious of which the Numerators and Denominators confift
of any Number of Simple Factors in Arith. Progreſſion.
k
"
a
a
a
•
&c. bx
+
105. (F)
(E)
p.q.r. Eic.
k+1
a+m.a+m. &c. cx
+
p+m.q+m, r+m. &c..
k+21
a+2m .a+2m &c. dx
•
p+2m.q+2m.r+2m. &c.
+ &c.-
"
AP + BQ + CR + DS+ &c.
1. Here a, a, a, &c. denote any uneven whole numbers; p, q, ", &c. any whole
numbers whatever.
2. P –
b x
+
CX
dx²² +21
+
+ &c.
Q=
P
b x
k
p+m
p + 2 m
k
k
CX
b x
сх
+
+ &c. R =
+
+ &c.
9
9 + m
r
r + m
&c.
"
A =
a-p.
p.a-pa-p. &c.
B
=
9. a-q. &c.
q—p • r—p.s—p. &c.
2
Sac.
p-q. r - q S
•
9
&c.
2
•
3. The number of factors in the numerator of each term must be less than the
number of factors in the denominator.
+ &c.
Hence the fums of the following feries are derived.
k
k+l
CX
+
+
dix²+21
p+2m.q+2m.r+2m. &c.
b x
p.q.r. &c. p+m.q+m.r+m.&c.
AP + BQ + CR + &c.
106. (F)
(E)
I
Here A =
B =
9-por-pos—p. &c.
,
I
p—q • r — q • s— q. &c. '
&c.
107. (F)
*
O F
163
SERIE S.
107. (F)
k+!
+
CX
p+m.p+2m. p+3m. &c.
+ &c.
k
b x
p.p+m.p+2m. &c.
dx²+21
p+2m.p+3m.p+4m. &c.
7- I
P―n Q + n X
R-nx
2
NI
2
n-2
S, &c.
(E)
3
I. 2
•
3.4.5. &c. n
12
m
•
I
Here n the number of factors in the denominator of each term.
XXIX.
+
Series of the Powers of a Binomial, the Terms of which are reſpectively multiplied into the
Terms of the Series
12
n
1,
>
X
g
r
n+ I
r+i.
23
1
X
X
r+1
n+2
r+2
&c.
108. (F) I±
n. px
+
n.n+i.p.p+1.x²
+
n.n+1.n+2.p. p + 1.p+2.2³
&c.
I go
•
I.2 r r+I
•
I.2.3.r.r+1.r+2
pq x
X : 1 ±
+
+
(ε)
I
↑
IFX
q. q- I. p. p+I.±²
I.2.r r+ 1.1 = x 1°
r+I.I+x
9 · 9−1 · 9 — 2.p.p+1.p+ 2 . +³
1. 2. 3.r.r+I
3.r.r+I.r+2.1=x
S
+ &c..
1. When the quantity n−r (= q) is a whole poſitive number the formula (2)
terminates in q-1 terms; and therefore in fuch cafes, the exact value of the infinite
feries is obtained. When q is negative, or a fraction, (x) is only an approximation.
9- I
2. The values of r, p, and x, may be any quantities taken at pleaſure.
3, The upper figns are ufed when they are affirmative in the propofed feries, and
the lower when negative.
Hence
164
THE SUMMATION
109. (F) I±
(E)
I
Hence are derived the following feries.
Cafe 1.
p =
P
I.
nx
|=
+
±
n.nti.x² n.n+1.n+2.x²
+ &c.
r.rti
r.rti.r+. 2
+1
9.x
2
9·9 - 1.1.2
+
J
r+ 1 • I=x
9.9-1.9-2.x³
r.rti.r+2.1 = x*
+ &c.
Cafe 2.
p = n, r = I.
N
2
2
2
110. (F) 1 ±
(E)
I
1x
n.n-3. n²
n
1
72
I
XI±
I. n².
I. 4.9
N. N - I
n
nt I
n
n+
n+21
x x² +
Xx³ + &c.
Ι
4
4
9
X
n . n − 1 . n² — I
X
+
X
I
IFX
1 •
4
IFX
4
3
n.n-
4.222
I.n² - 4.n²
+4
X
+
X
&c.
3
+*
I
•
4.9.16
4
IFX
Put p =
응​,
n
n =
and
,
Cafe 3.
n-r
a whole affirmative number.
m
m
M
n
I±
xx+
n n+m p p + v
X 2²±
n n+m n+2m
γ r + m
V
20
r r+m r+2m
III. (F)
p+o
r ບ
p+20
x x³ + &c.
20
30
+
ข
r.r+m.v
(E) 1±
n. px
g
•
n.n+m.p.p+v.x²
•
20
It
n.n+m.n+2m.p.ptv.p+2v.x³
r.r+m.r+ 2m . v . 2v. Zv
+&c.
XXX.
OF SERIES.
165
XXX.
Series of the preceding Form where the Quantities n-r, and v, are whole pofitive
112. (F) 1± x x +
(E)
bx
IFX
n. p
r.v
I-V
дета
+
IFX
Numbers.
n. n + 1. p.p÷1
r.r +1.0.0+I
Xx²±
n.n+1.n+2.p.p+1.p+2
T. +1.+2.v.v+1.0+2
xx³ +&c.
n―r.p―v+I X
X
+
r-v+1. I
•
1. 2
&c.
I-V
1. Here h±
n—r.n—r—1.p¬v+1.p-v+2 X
r−v+ı.r-v+2
n—r.n—r—I.n—r—2.p¬v+ 1.p−v+2.p¬v+3 X
r-v+1.r¬v+2.r-v+3.1.2.3
till it terminates,
x3
IF
n−v+1.p¬v+I n-v+1.n-v+2.p-v+1.p-v+2
r−v+1.r−v+2.1.2
Xx+
r-v+I. I
I.r-v+I
2.1-v+2
X
X
3.r-v+3
N-V+1.p¬V+ 2 n-vi+2.p-0+2 n-v+3⋅p-v+3
continued to v— 1 factors.
2. When the figns of the propofed feries are + and
fign is uſed; and in this cafe v muſt be an even number.
xx²,&c
x &c.
alternately, the negative
Hence is deduced the fummation of the following feries.
113. (F) 1 ±
n.p
n.n+m.p.p+w
xx+
r.v
r.r+m.v.v+w
n.n+m.n+2m.p.p+w.p+ 2w
r.r+m.r+2m.v.v+w.v+2w
(x)
hx
20 บ
LU
| 3
W
bx
พ
૩
+
મ
X
X
+
5. บ
1FX
q. q— m. t .t+w
s.stm.w.2w
+2
X
IFA
9·9—m. q—2m.t.t+w.t+2w
s.stm.s+2m.w 2w. 3w
till it terminates,
xx³, &c.
>
*.3
X
&c.
-V
k. t
k.k+m.t.t+w
X : I +
xx+
s.w
S stm.w. 2w
+
kik+m.k+2m.t.t+w.t+2w
s.s+m.s+2m.w.2w.3w
if it does not terminate fooner.
V
W
× ×³, &c. to
terms,
W
1. Here
166
THE SUMMATION
V
n-r
1. Here
and
>
,
are whole pofitive numbers.
W
In
2. The quantities n, r, p, v, are put =
n
р
ย
reſpectively; alſo
in
m
ω
น
M V
บท
U S
n-r = 9, r+m
=s, p-v+ w = i, n+m
ย
k, and b = ±
พ
พ
X
270 s+in
t+w.k+m
3w.s+2m
U-W
X
X &c. continued to
t +2w.k+2 m
t . k
fators; the negative,
10
lign obtaining when the figns of the propofed feries are alternately and, and
ปี
20
at the fame time an even number.
XXXI.
Series arifing from the expanding of a Binomial Surd.
114. (F) x
#1
72
or, x
(E)
771
+
พ
m
n
A
y
+
m
n
y
m-2n
B +
y
m
C +
·3n
D
γ
+ &c.
X
2 n
312
X:1+ X +-
X
172
m n
X
X
n
2 ከ
m
x+y!
+
m
n
X
n
2n
4 n
X
Here A, B, C, &c. denote the preceding terms.
Hence are derived the following ferics.
n-an
3 n
2n
درو
X +&c.
nx
115. (F) a² × : 1+
n + I
A +
n+2
x
X
B+
X
C+ &c.
a+x
2
a+x
3
a+x
(E)
a+x
n
2.
Z
116. (F) p ±
2p
8p³
3
16ps
525
128p
3
+ &c.
I
(E)
p p =± z z
.
B3
B³
5B'
117. (F) I
&c.
3
9
81
3
(E)
118. (F) 1 +
(E)
X X
m²
2
2 m+
VI - B³.
6
+
2 mo
&c.
m² + x²
119. (F)
OF SERIES.
167
vn
119. (F) √n × : n + å y
11 y2
19 y³
&c.
32 n
128 n²
4
(E)
nty x vn-y.
2
5
2
120. (F) r´ × : I
2x+
6x6
27 +8
5 r²
&c.
25 r
1257°
625r
S
(E)
V p²
And in like manner may be derived infinite feries ad libitum, with finite expreffions for
their fums.
XXXII.
Series derived from Divifion.
H
Z4
zs
+ + &c.
121. (F) ≈±
(E)
122. (F)
(E)
m²
n
H
a
+
m² x
n
a
2
a z
az
m² 2:2
m² x3
+
+-
+ &c.
n³
n+
771 172
1 F x
TO
123. (F) 1 ± q² + v4 ± 0° tos ±oto + v²² ± &c.
(E)
I
I F V
I
3
124. (F) 2x
2x + 7 =
13² + 34
&c.
I : 2
3:2
(2)
2 π
2 + 1
I : 2
- 3 T
༡༤
125. (F) I
|
52³
+
a²
12 23
+ Sic.
(E)
a²
a²+zaz. Σ
This order of feries, it is obvious, may be continued fine fine.
Bb
XXXIII.
168
THE SUMMATION
}
XXXIII.
Series of the Reciprocals of certain Progreſſions, and their Powers.-
I
{
126. (F) ;+1.5-1 +
I
2r+ 1.2r— I
£
I
+
+ &c.
3r+1.3r− I
(E)
127. (F)
(E)
I. r
I
r+I.r— I
HIN
Nπ
+
I
2″ + 1 .2r — I
플​.
2
4
I
+
+ &c..
3r+1 · 3r~ I
128. (F)
(E)
I
r+i.r-
r - I
HIN
Ι
+
2r+ 1°. 2r — 1
Ι
+
3
+ &c.
3
3r+ I
3r-I
플
​3 n π
4³
Where 2 is the periphery of a circle, radius 1.
value of r, viz. If r = 2, 33, 35,
reſpectively.
2
And n is determined from the
or, &c. n will be 1, 3, 5, or 7, &c.
Hence ariſe the following feries.
Cafe 1.
r = 2, n = 1.
129. (F)
(E)
130. (F)
(E)
I
1.3
I
12.32
I
+
+
I
I
I
I
+
+
+
+ &c.
3.5
5.7
7.9
9.11
1.
I
3.5°
4.
I
2
+
HIN
.
I
I
+
+ &c.
5.7
72.9°
I
131. (F) 2.3, +353 +5.73 + 71.91 + &c.
(=)
33
플
​1
3.2
3
Cafe 2.
OF SERIE Ș.
S.
169
Cafe 2.
5•
4 = 2/3, n = 5.
132. (F)
(E)
133. (F)
I
I. II
I
2.5°
I
12. I I²
I
(2) -
I
+
+
+
3. 13
I
3.7
I
I
+
+ &c.
5.15
+음​.
I
+
32.132 5.15
I
2.5+ 3.7
1
I
+
I
9
+
I
2
4°
134. (F) 13.11³ +3.13 ± 5.15
3
3
+ &c.
•
+ &c.
I
I
(2) 2.5°
+
I
+
3
3.75 93
3π²
4³ • 5+
And by affuming different values for r, may a variety of other feries be derived.
XXXIV.
Other certain Series the Summations of which are exhibited by the Periphery of a Circle.
135. I
I
136. 1 +
137. I
138. 1 +
I
139. I +
I
+
3
I
9
I
+
+
I
5
25
I
!
-| -| -|o mlio w/2
-la ala alão Hico Him
3³
I
+
5³
+
I
I
3 5'
(F)
+
+
(E)
મા
I
7
+
9
&c.
I
+ &c.
81
8
I
+ &c.
3
16
4
3.32
H
+
||- / / $1
7
+
+
-15% ala ala ala
I
+ &c.
+
I
I
II 13
ļ
&c.
212
Bb 2
140.
170
THE SUMMATION
140. I +
141. 1 +
+ 3/2
Nolm N/A N/
+
w/w w/n wilm
(F)
+
+
+
I
4
-/+ =1+
I
+ &c.
+ &c.
4
I
+ &c.
(E).
6
4
3.2.3.5
142. I +
143. I+
20
+
I
2
+
2
3
I 3
2·4·5²
·
+
1.3.5
2.4.6.7
+ &c.
2
5
7
II 13 17
144.
&c.
3
6
6
10 14 18.
145.
146.
3
147.
m/+ 0/5 m/m
3
5
7
II
&C..
•
4
4
8
I2
5
7
I I
&c.
•
4
8
10
2 6
29/0
5 7
It
13
17
19
&c.
8 10 14
16 18
22
148.
32 52
62
&c.
·
3
2.4 4.6
6.8
22
42
62
82
149.
&c.
•
1.3 3.5
5.7 7.9
42 82
122
162
150.
&c.
3.5 7.9
1 I
13
15.17
33
1513 151
3° ± 1 5' 1
53
73
113
I༣3
&c.
3
7³± 1
3
11³+
13³ — I
945-
L.2XT
元
​4
2
75.
T
2√2
元
​6°
2
π
4√2
= W/
32
30
XXXV.
OF SERIE S.
171
XXXV.
52
152. (F) s +
2·3r²
4 • 5r²
A + 3+ B + 5+ C + 777
Series of which the Summations are expreſſed by the fame Arch of a given Circle.
35
-2
55
2
6.722 8.9r²
75
D+ &c.
↑ -C
g3
c3
153. (F)
3
+
+
+
I
2.3r²
3.5.r
2.4.54 2.4.6.7°
+ &c.
154. (F) Varo +
155. (F) t
156. (F)
T
-
24
ข
32 v
A +
3.4r
5.8 r
B+
520
C + &c.
7.127
t7
+
3 r²
31
5 rt
+
ō t
&c.
7 2°
9 r
3
+
373
5 x5
+ &c.
7x7
S-r
157. (F) r ×:
S
+
3.Ss
دمو .
+
2.3S³ 2.4.555
+ 3.5.S-77
+ &c.
2.4.6.7S'
158. (F)
+235 +
заб
2.303 2.4.5 as
+
3.578
2.4.6.707
+ &c..
(E)
Circ. arc. rad. r.
1. The fine, cofine, verfed fine, tangent, cotangent, fecant, and cofecant, are
denoted by s, c, v, t, T, S, and a, refpectively.
2. A great variety of thefe fpecies of feries, where the fummations are exhibited
by the fine, cofine, &c. or multiples, of a given arch, may be found in feveral trea-
tifes on trigonometry, as Emerfon's, Mafere's, Mauduit's, &c. Mr. Hutton has
alfo given feveral new feries for rectifying the circ. arc. of 45°; which, for their
fimplicity and quickneſs of convergency, are far fuperior to any others that have ye:
been investigated. See Philofophical Tranf. Vol. LXVI. Art. 28.
XXXVI.
Series of the Powers of a Quantity, the Exponents of which are in a decreafing Arith.
Prog. and the Terms refpectively multiplied by the indefinite Quantities a, B, Y, &c..
2
159. (F) α x² + B +
2- -I
*
x
+ y²-2+d+3+€+²-4+ &c.
2—2. +
(S) A +*+¹ + B** + C +˜¯ + D +²−² + Ex
+*+ −
+B+C++ Ex-3+ &c.
1. Here
172
THE SUMMATION
α
B
1. Here A =
-, B =
Z+I
A A,
Z A
z+I.A
2
Y
Z+I 2
•
C=
BA-
Α Δ',
Z-I.A
2
2.3
Z-I
א
Z
Z-I
D=
CA
BA²
z+1.2.Z — I
Ꭺ
Α Δ',
2-2.4
2
2.3
2
•
E
Z 2
E =
DA-
2— I. 2 – 2
2CA²
Z.Z-
3.4
1.2-2
BA3
2-3.A
2
2.3
2.3.4
&c.
2+1.2.2-1.Z ~ 2 Α Δ',
2
•
3.4.5
2. The fum arifing by fubftitution in the formula (S) will generally need a cor-
rection; which may be effected by making z = o, and ſubtracting the refult from
the faid fum.
3.
The factors in the propofed feries muſt be in the fame arith. progreffion.
XXXVII.
Series of the Powers of a Quantity, the Terms of which are refpe&ively divided by the
Terms of an Arith. Progreffion.
9.3
got
160. (F)
+
+
+
+ &c.
n+d
n + 2 d
n+3d
n+4d
↑
dr
+
A +
r—I. V
r-1.vtd
2 dr
r-1.v+2d
B+
3 dr
r-1.0+3d
C+ &c.
(S)
go
dr
2 d r
+
A +
C+ &c.
n+3d
r-In+d ↑ - 1.n+2d
1. Here vn+dz; and A, B, C, &c. denote the preceding terms in each
leries.
2. This formula gives the fum of 2-1 terms of the propofed feries.
3. If r be less than 1 the latter feries gives the fum of all the terms ad infinitum.
XXXVIII.
O F
173
SERIE S.
XXXVIII.
Series of the Powers of a Quantity, the Terms of which are refpectively divided by the
Terms of a Series confifling of two Factors in Arith. Prog. where the Com. Diff. is
Unity.
161. (F)
111
m²
#3
M4
+
+
+
+ &c.
I. n + I
2.8+2 3.7+3
4.7+4
Z
m
I
Q+
m-i
2 Am
n-
I
X:
3 Bm + n
1. n − 2
==+
+
+&c.
m-1.2.2+I
(S)
m X:
I
+
m-1.v.vt I
(E) Q+ m² X:
1—m.v.V+1
m−1.2.2+1.Z † 2
2 Am--n- Į
I
771- I
ท I .V. U+1.0+ 2
m-1.2.2+1.2≈+2.2+3
+ &c.
I
2 A m
+
1 — 771
•
7- I
V.U+I.0+ 2
+ &c.
I
2 A m
n-
I
1. A =
=
B:
=
>
C =
,
771
I
m- I
4 Cm
D=
,
n − 1 . n − 2. n−3
7- I
3 Bm + n − 1. n — 2
m— I
&c. Q= the fum of 1 initial terms of
the feries; which must be found by addition.
z
-I
2. The formula (S) gives the fummatrix for x-1 terms; and (2) gives the fum
ad infinitum, when mis lefs than 1.
XXXIX.
Series of the Compounds of certain Arith. and Geom. Progreffions.
S
162. (F) *
rta.n
x:1+
·mx" +
sosti
t.t+I
m² x
s.sti.s+ 2
27
+
m³ x3 + &c.
t.t+l. t +2
rta.n
(E)
X
2
r.rti.r+2 (a)
m² 9·9-1.9-2 (a)
12
I-mx
X
rn
7
}}
$ 12———— I
I-
× : A − Am x” + A m² x
–
27
&c. (a).
Where qsr+1, (t being = r + a + 1) a = o, or any pofitive integer;
""
"
I
A =
A =
>
nr
a-s.nAte
n.rti
#
a-s- -i.”A+b
Α A =
A
,
n. r+2
a—s—2.nA + c
n. r+3
Stc.
>
a
Q- I
a
a — 1. Q-
2
in which b =
C=
&c.
>
>
2
2.3
Hence
174
THE SUMMATION
MATION
Hence are deduced the following formulæ.
x: 1 +
t
"
Mx" if
9
Cafe I
y.rtl
t +I
I.
211
m² x
163. (F) xta.n
r+a.n
(E)
+
X
a
r.rtl.r+2(a)
1.2.3
(a)
m
X
I
m x
rn
"1
+ &c.
S
x
I - m x
"
x
#
x" x: A - A mx" + &c. (a).
* X:
The upper or lower fign obtaining, according as a is even or odd.
Cafe 2.
Make q
1, and for m write m.
164. (F) x
rta.n
r
X: I
m x
+
t
r.rtı
t.t+I
m² x
2.11
&c.
r+a.n
(E)
X
r.r+1. r + 2 (a)
I + m x
X
77L
a
I. 2
3
(a)
13
a
ر
† MI
I+mx"
"
X × : A+ A mx”+A m²x² + &c. (a).
r.rti.r+2.1+3
1.1 + I. {+2.1+3
Fr
1724 444 ± &c.
Hence alfo arifes,
165. (F) x
rfa.n
•
X:+
m² x²n +
1. (E) i+a.n
I.
X
a
r.r +1.r + 2(a)
1.2.3 (a)
a
rЯ-I
27
I + m x
2
a
n
1-I
X
I mx
I+mx"
n
n
2
I-mx
X
***
A + A m² x
+ A m² 4"
or, if a be odd,
#
-
VI
(1/2)
Amx”+A+m³ x3″ + Am³ x5"
A m³ x 5" (a = 1 )
2
2
2. (E)
1
OF SERIES.
175
+
2. (E) ±
rta.n
та
r.r + 1.r + 2(a)
X
X
I. 2.3 (a)
rn
rn-I
+5 1
X &
+
X
2n
I + m² x
C
2
A- A m² x²² + &c.
&c. (-—-—-)
or, if a be odd,
2
ix
rn+n-1
I + m²
X
2/1
X
"
#
Amx” — Ä m³ x³″ + &c. (ª—¹).
I
- A
A
2
1. Here S and C denote reſpectively the fine and cofine of a times the circ. arc.
rad. 1, tang. mx”, and fecant §.
2. When the figns of the propoſed ſeries are affirmative, the formula 1. (E) ob-
tains; if alternately changeable from to, the formula 2. (E) gives the fum.
3. If a be an even number the upper figns and quantities are ufed; if odd, the
lower. And in the former cafe, the initial fignor
Q
is uſed in the formula
2. (E) according as — is even or odd; and in the latter caſe, as
2
a- I
is even or odd.
2
XL.
Series arifing from the Binomial Theorem.
(ε)
I
166. (F) 1+px +
p.p-
+
2
p. p — 1. p—2
2.3
x² + &c.
I+ *
167. (F)
(E)
I
Hence are derived the following feries.
1.2.3(p+1)
p+2
+
x²+3
2·3·4p+1) 3.4.5 (+1)
&c.
I
I
x: 1+ *² × L.I+x-x
1.2.3 (P)
Р 24
p-1.p-2
p.p-2
p.p-1
2.3
p-1.p-2.p-3)
p.p-2.p-3
I
**
&c.
p.p1.p-32·3·4
p.p−1 . p-2
168. (F) 1+ 2
p² .p - p²
p² · p -
I
+² +
+
· P-4² + &c.
2
•
2 · 3 · 4
2.3.4.5.6
(E)
/1 + x + x +
1+*
x
2
Сс
1
XLI.
176
THE SUMMATION
I. (E)
3. (2)
1.3.5.7 (-1/2)
72
n.n+2.n+41 )
2
1.3.5.7 (
(음​)
n.n+2.n+4(
I
1 . 3 . 5 (21)
n
n+1.n+3(
n + 1. n + 3 ( 2 )
2
X
XLI.
Other Series deduced from the Binomial Theorem.
169. (F) x
naf z
I
X:
n+1.n+2
22 x
n+2.n+3.+4
2.3.26
I
F
n+4. n + 5
•
6
n+6.n+7.n+8
+
1.2.24 +4
n+3•n+4.n+5₁n+6
+ &c.
4n
S
1+x
n²-22.n²-42
n
-22
n
2
n² 2.3.4
n².n²-42
n².n²-2²
2.3.4.5. 2
I+x-x
2
S
X:
Xcirc.arc.rad.1,fin.x +
n
212
2
x² n²-22
n²
+ (2/-).
2.3.4
X
-Ħ
x
X
4 n
S
x
2
X
n²-1.n²-3²
I
25
n+I
x:x+
~ + 2n².n² - 3²
2n
21² 12.3
2n².n²-1
2.3.4.5 2
1.3.5
(==)
2
C
+.(E)±.
X:
2
S
I
n²-I
X:*
+
213
I
2n² 2.3
&c.
&c. ("+
2
("+").
n + 1. n +3 (2-—- ¹)
1. When n is an even pofitive number, the formula 1. (E) gives the fum of the
feries, if the figns be alternately + and
But when the figns are +, the formula
2. (E) gives the fum; S being the fine of n times that circ. arc. and the initial fign
isor, according as
n + 2
2
is even or odd.
2. If n be an odd pofitive number the formula 3. (E) exhibits the fum when the
figns are + and —. If the figns be +, 4. (E) is the formula to be uſed; C being
he cofine of the circ. arc. whereof S is the fine. And the initial fign is + or
according as
n + I
is even or odd.
2
lf
OF SERIES.
177
If no, we have,
1. 2. 3. 26. *³
2.3.4 3.4.5.6 4.5.6.7.8
22. x4
170. (F)
1.2
1
•
2.24.xo
+
+
+
(E)
플 ​X
S
x
X
+ &c.
XLII.
Series of which the Terms confift of an indefinite Number of Factors in Arith. Progreſſion;
each Factor being divided by the corresponding Term of the Series of Natural Numbers,
beginning from Unity.
n-I 1-2
n-zti
171. (F) n.
+ m x n
2
3
1- I
2
n-2
3
Z-I
+m.
M-I
2
n. I
n-2
xn.
2
3
2-2
111 I 972-2
I
72
m
Xn.
n-x+3+
2 7-x+4
+ &c.
2
3
2
3
Z-3
M-I M- 2 m-3
M-Z+I
to m.
2
3
4
Z
n+m-1
n+m-2
(S) n+mX
n+m-3
n+m−x+I
2
3
4
א
Z
XLIII.
Logarithmic Series.
172. (F) a³× L.±a+dXL.a+d+a+2dXL.a+2d±a+3a² x L.a+31+ &c
1.(S)
I
Satzd
.L.a+zd a+zd². L.a+zd
p+i.dl_apti. L. a
+
-p-I
pAdCatza"
2
1
pti..
d
.L.a + zd
p-I. L. a
플
​-a. L. a
P.L.
&c.
a+zd² +¹ Ád fa+zd²-1 p-1.p-2
+
2
ap-I
Σ
+ p.p-2}X
Ad³ fa+zd
Cc 2
p.p-1.
Wo
2.3.4 -QP-3
&c.
2. (S)
T
178
SUM
THE SUMMATION
2 (S)
2
2
14
a+2zd². L.a+2zd
-ap.L.a
+
-p-1
-p-I
2² - 1. p Áds a a+2zd´´.L.a+2zd
- [a
² — 1. Á d√ a + 2 z d¹-z p-1.p-2
2
·Cip-I
+ p.p-2 X
P.P-I
2
--1.L.a
P-3
}
2-1. Ad³ a+2z d² = 3
2.3.4
V
-at-3
I
1. Here Á =
A
A =
I
I
2
A =
A =
A =
>
"
6'
30
42
30
denominators of the laft terms of the feries denoted by B, D, F,
&c.
&c.
56, &c. being the
66'
&c. in Order III.
2. When the figns of the feries are affirmative, the formula 1. (S) gives the fum
of terms; when they are alternately changeable from to, the latter formula
+
is ufed, which gives the fum of 2 z terms, or z initial differences.
x
173. (F) a+¿³ × L.a+d+a+3a². L.a+za+a+5a². L.a+5d + &c.
a + 2 zd .L.a+2zd pАdfa+2zd
I
2. p+i.al
p+s
-apti. L.a
a+2zd¹¹.L.a+2xd
2
&c.
l-ap-:
— ap-i. L. a
S)
I
a + z zd
Ada+2xd
2 x dp - x
2.p+1
d
pti
2
p-I
a
a
p.p-I
-p-x p-1.p-2 2³-1. Ad³ √ a+2zd³-3
—
3
√
p.p-2x-
2.3.4
3·4-ap-3
From theſe general logarithmic feries may a great variety of particular feries be de-
rived by taking p = 0, 1, 2,
O, 1, 2, or 3, &c.
XLIV.
174. (F) a" + B" + x
m
+ d + " + &c.
Series of the Powers of the Roots of Adfected Equations.
+ B + 2"
m
p” — m q p
pm-² + mr p
m-3
+mt
ms
m-4
m-5
m -3
P
+m.
92
2
m
•
1-4.gr
-m v
(S) +m.m m- 5.9 s
112
+m.
Here
m 5 m - 4
2
mn
5
2
2
3
+ m w
m.m-6.qt
3
in-6
か
​+m.m
m-6
5.
β
m-7 &c.
q² r
2
m
- m
6.rs
ß, Y, d, ɛ, &c. are the roots of the equation
ад
71-I
1-2
x
px
11-3
1-4
+9x
r x
+ sx
tx²-5+v x
vx
n 6
&c.
175.
&c.
O F
179
SERIES.
(E)
175. (F) a + b + c + d + e + &c.
p-29+3r-45+51-&c.
1-p+q-r+s-1+ &c.
x+I
p-p
2-p-x+1.p +zp²
Z-I
-9X
2
+rx3¬² p−x+i.p
24-
+zp
I-P
sx 4-3p-z+1.p
2. 3
-2
Z -2
2—1
(S)
+ q² × 4-2p-x+1×z−2. p
+2
2.1-p
4%-4.p
Z. Z-3.p
3
+ t × 5− 4 p −z + 1. p
+xp
3
2
4.
Z
·3-2.2-4 · Þ
-grx5-3p-2 + 1x - 3⋅ p
&c.
Here a the fum of the roots of the
the fum of the cubes, d
the fquares,
fame roots.
+ + 2 ≈ ² — 6 ≈ — 5: P
I-R
above adfected equation, the fum of
the fum of the biquadrates, &c. of the
a
176. (F) BB 2c did s
(S)
α
A-B+IX 2 C
દ
&c. + ab Ba je dd & &c. to m factors.
ва з
I 2.3D+1.2.3.4 E
•
+ 1 . 1 B B — I
•
1.2 B C
1. 2. 3. 4. 5 F +1.2.3.4. 5. 6 C
+ I. I. 2 3 B D - 1. 1. 2. 3. 4 B E
2 BBC
•
BBB+
+ 1. 2. 1. 2 C C − 1.2.1.2.3 C D
1. Here A = Sª . S². Sˆ.Sª. &c. B =
sa+b+c. A
Sa. So. Sc
so.sc
D=
b
ga+b+c+d A Satotit. A
Sª.56.5°.5¢ ga.s³.Sˆ .S¢
Satotitate. A sa+b+c+d+f.A
+
sa+b+i. A
+
sa.sh.
Su. Sb. St
S.Sc.
Satb. A
Sa.So
sate+d. A
Sa
sate. A
+
+
Sª.Sc
Satd. A
Sa.sd
+ &c.
+ &c.
+
+&c. BB-
+
b
E =
+
+&c. BC=
S".S..S".S' Sª.S". S°.S¢.§ƒ
d.St
sa. Sb.Sc.so.5°
Satb setd. A sete goto A
S².5°.S.S¢ S.5°.Sc.5¢
goth getate. A sate gate+f. A
+
+ &c.
S“.5°.8%.5° ƒ+&c.
भ
180
ATION
THE SUMMATION
SUM MA
ga+b+c+d+c+f. A
F =
+&c.BD=
ა
Sª.5o.Sc. sd.se.sf
sa+b+c sdte+f. A
gatate+d getf. A satutate satƒ. A
+
5.5%.Sc.S« . ° .Sft sa .st .Sc.s" .sf
Sa+b+d getetf. A
CC=
+
+ &c.
Sª.Sb.Sc.5ª .S°.Sƒ
Sª.sb.Sc.Sd.se .sf
Satbsctd setf. A
BBB =
+
S.S.S.S.S.S
Sate_sota se+f. A
Sª.So .Sc.S«. §*.sf
+ &c.
+ &c.
Where Sª = aª + Bª + &c. S² = a² + ß² + &c. Sˆ = aˆ + B² + &c.
Sa+b
atb
= a + &c. Sate
sa+b+c a+b+c
= aª+ " + &c. satd a+d
α + &c.
= a + &c. Sa+b+d
&c.
a+b+d
= α
+ &c.
2. The quantities a, B, y, &c. are as before; and a, b, c, &c. given indices.
3. The number of factors (m) in each term muft not be greater than n.
XLV.
Series of Compound Radicals.
n
11
17
177. (F)
(E)
a +
a + √ a dic.
Root of the equat. x
n
11
178. (F)
(E)
m
a
n
n
n
a
a
a &c.
ガー
​= a
Root of the equat. x
179. (F) Va+b
171
Na+b
m
a + &c.
(E) Root of the equat. x
m
180. (F) √a+b
n
c+d
m
Na+b
&c.
n I
-ax = b.
a.
n
171
172
(E) Root of the equat. x
a Xcx + d x x
a
- b" x.
181.
OF SERIES.
181
#1
181. (F)
√√
b +
a
b +
(z) Root of the equat. x
To
b + &c.
√
+ bx* = a.
n
182. (F) Va+b
c+d
Natb
c+d
a + &c.
(E) Root of the equat. cx
+dx"
**+ ac+b.*+ a
x+ad = 0.
XLVI.
Series expreffed by their General Terms.
2 z−1.2 2.2z+2.
183. (F)
(S)
6x-2
3
× Z.Z+1.2+2.
42+6
184. (F)
(E)
185. (F)
(S)
186. (F)
(S)
z. 2+1.z+2. z+3
z.x+4
34×5
e1/m
x+1.x+3
4Z+I
2x−1.22.2z+1.22 + 2
2.2z+
2.2z+1.2x+2
z2
32−2·3~—1. 32.3≈+1.3%+2
12
·
2. z+1
3≈+1.3%+2
187.
182
THE SUMMATION
187. (F)
(S)
3
a+4a² + a³
a.
I
a
22
z2
23
3 a z²
Ita
@+4a² + a³
+
3
хзач-
a
a- I
a
a-
a
a-1 .a
a-1 .a
2
+
188. (F)
(S)
Z
•
Z+I
2.3
27
8
I
4.3
2-I
× ≈ +2.≈+2 + 1/2•
189. (F)
(E)
(S) -
Z.Z+I
2+2x+3·4
•069243•
22
1
I
4
I
X :I
A
ซ
V
3.4
3.U+I
6
2
3
B-
C, &c.
+
ข
3.0+2
3.0+3
1. Here A, B, C, &c. denote the preceding terms; and v = x+3•
2. If in fubftituting the natural numbers 1, 2, 3, &c. fucceffively for z in the
general terms, in order to diſcover the form of the feries, the firft, &c. terms arifing
ſhould not coincide with the propofed feries, thofe fuperfluous terms must be ſub-
tracted from the formula (S); which will thereby be properly correaed.
3. By the fame method the fummations of theſe feries have been derived, may a
great variety of others be found, viz. by taking the correct integral of the z+ ge-
neral term of the propofed feries.
- Ith
XLVII.
OF SERIES.
183
XLVII.
Series of the Powers of the Natural Numbers, the Terms of which are reſpectively dividea
by thofe of a Geam. Prog. the Ratio being equal the first Term.
I
!90. (F) +
g
"
2
+
3"
+
4
= + &c.
i
I
(E)
+u
X
r
r
+
X 2
I
X 3″-2” X
n
nti
+
nt I
+
I
n+ I
I 2
+
•
n
n+ I
n+I 12
·
nti.n
+2" X
I
I.2
I.2
•
n-I
3
+ &c.
to n terms.
- z" p P — n z" ~ ' pQ
+ 1-1 X Σ.
Here p =
Χ
1
2
r
n.n-I
n- 2
X:
þR.
I. 2
n—1.N— 2
1.2.3
x x"−3 pS - &c.
; and P, Q, R, &c. are the reſults of (E), by writing for n, o, I.
2, 3, &c. respectively.
XLVIII.
Series of a decreafing Progreſſion of Numbers, the Terms of which are respectively multi-
plied into thofe of a Series of Powers.
3
191. (F) a − bx + c x² - d x³ + e x² − ƒx³ + &c.
(E) a
b x
I+*
Ú x²
"//
D x*
4
&c.
I + x
I + *
I + x
1. The quantities a, b, c, &c. muft be a decreafing progreffion of numbers, and
be fo related that their firft, fecond, third, &c. differences fhall alfo each form a de-
creafing progreffion.
2. The quantity a muft not be greater than unity.
3.
Ď, ´, Ï, &c. denote the firſt differences of the feveral fucceffive orders of dif-
ferences, viz. Ó = b−c, Ď = b −2c+d, Ï − b−3c+3d−e, D) = b−4c+6«
V
−4e+ƒ, D = b−5c+ 10 d−10e+5f−g, &c.
D d
IV
XLIX.
184
THE SUMMATION
XLIX.
Permutative Series.
192. (F) abc(x) + acb (x) + bac (z) +bca(z) + cab (z) + cba (≈) + &c.
(S)
a+b+c+ (z)
Z
X 1.2. 3 (≈) X : 1+10+100+ 1000+ &c. (z).
Here a, b, c, &c. denote reſpectively the figures of which the propofed number is
compofed, confidered as fo many units; and the terms a, b, c, (z), &c. are not taken
as the products of the quantities a, b, c, &c. but as the aggregate of the numbers
refpectively denoted by thoſe letters, according to their local value in each term.
L.
The continual Products (to any Number of Terms) of a Series of Fractions, of which the
Numerators and Denominators are in Arith. Progreſſion.
a
193. (F) =—-/- × X
a+r a+2r a+3r
X
X &c.
b + r 6+2r b+3r
(S) The number correfponding to the com. log. which is expreffed by
B
X
C
कर
B
C D
A x log.x+PX: + + &c. — A x log.b—P × : + + + &c.
b
62
63
Where P43429448, and x = b+xr. Alfo A = 1, B =
r
n-r
2 r
n, с =
2
3 nr-2 n².
3nr-2n
2
n-
nr
n, D =
12 r
n², E
3.4r
24
4.5r
n4
3 B-42C+6r D
12 +
F
4
Ins
3
got B + 5 r³ C — 10 r² D + 10 r E
доб
no
n +
G
n
+
5.6r
5
6.7r
25
6
ys B − 6 r^ C + 15 r³ D-20 r² E+15rF, &c. n being a-b.
LI.
Series deduced from the Binomial Theorem, being the Powers of certain Arith. Progreffions,
and their Reciprocals.
+I
+
X
↑
go a grĬ
↑ . r-I . r=2
194. (F)
---
nx
1-2
X
n³ x
&c.
rti.n
2
2.3
2.3.4
(ε)
*.
Hence
OF SERIES.
185
Hence are derived the following feries.
195. (F) m² + m+n + m+2n +m+3n + &c. (z).
I
m+zn
(S)
mrti
r. r — I. † - 2
2. Än³
2 ·
3.4
HIN
m+zn
+
1-m"
3
m+zn
+
Y-3
m
rÁ m+zn
21 m
r-I
-†----- I
+
r.r_1.r—2.r−3.r—4. Än³ (m+zn
2 · 3 · 4 · 5 · 6
196. (F) m+n+m+3n+m+5n+ &c. (z).
&c.
m
"
•I
I
m + 2 zn
rAn m+2zn
2.rti.n
2
YI
m
m
(S)
3
m + 2 z n
<
25 I
# 3
m
##
2.3.4
r— 1. r — 2.r−3.r−4. An³ ¦ m+2zn
2.3.4.5.6
197. (F) m² - m+n+m+2n -m+3n+ &c. (2x).
r
m +2 z n
(S) — 1
1
+
m'
I
2²-1.r.Án (m+2zn
"
2
1. Here Á = ½› A =
6
r-I
m
I
-I
+
772
2+- 1.r.r-1.r-2. An³ {m+2zn
I
#
A =
A =
30
42'
2.3.4
I
•
Scc.
n
r-3
&c.
r-3
- m
36, &c. ſee Order 3. N° 6. &c.
2. If r1, &c. we get the reciprocals of theſe feries.
LII.
Series of Fractions of which the Numerators and Denominators are composed of the Roots of
a given adfected Equation.
198. (F)
a − bx + cx² - dx³ + &c.
-
A−Bx + Cx² -Dx³ + &c.
+ &c.
(S)
a - ba+ ca² + da³ + &c. a - b B + c B' + dB³ + &c.
+
+ &c.
A-Ba+ Ca + Da + &c. A bB+CB +DB + &c.
1. The terms of the formula (S) muſt be reduced to a com. denominator, and the
values of p, q, †, &c. ſubſtituted.
2. The quantities a, B, y, &c. are the roots of the given equation ”
- px
+
7
&c.
3. a, b, c, Sic. A, B, C, &c. are given coefficients.
Dd 2
LIII.
186
THE SUMMATION
LIII.
The continual Product of an indefinite Number of Factors in Arith. Progreffion; from
whence are deduced the ultimate Values of certain Products confifting of an infinite Num-
ber of Factors.
199. (F)
(S)
r.r+1、r+2 . (z)
p+r+i.p+r+2.p+r+3.(x)
√ T-28 XX
I *
1 X Xx
22+271
2y--I
Where p+1 and r may be any pofitive numbers; and in this formula denotes
the whole fluent of the expreffion to which it is prefixed.
i
EXAM-
1
OF SERIES.
187
EXAMPLE S,
ILLUSTRATING THE DIFFERENT ORDERS OF SERIES.
I.
Required the fum of 100 initial terms of the feries
2 + 5 + 8 +11 + 14 + &c.
This feries correſponds with N° 1, where by compariſon we have a = 2, d = 3,
名
​% = 100;
× 2a + z-1.d = 50 × 4+99 × 3 = 15050, the required
2
fum.
Required the fum of 25 terms of the ſeries
ΤΣ
1 {
n + n→ 11b + n~b + n − b + n − b + n − {xb + n − b + &c.
1
This coincides with N° 2, from whence we have an, d=b, and ≈ = 25;
z
2
Z
1 × 2a-x-1. d = 25. π—
× 2a-x-1.d = 25. n−b.
145 +
If we fuppofe n
-b. If we fuppofe n = 150, and b
= 150, and b = 60, then
140 + 135+ &c. the fum of 25 terins
d = 5, and the ſeries becomes 150 +
of which is 25. 150-60 = 2250.
And
0
d
30%, the number of terms in the whole ferics; hence (x) =
15 X 300-29 × 5 = 2325.
2.
I
Suppoſe a bullet to fly 20 miles the firſt ſecond of time, 19 miles the fecond, 18 the
third, 1733 the fourth, &c. how far would it fly in 1 minute, alfo in 1 hour? and
bow far were it to proceed for ever?
Comparing this feries with N° 3, we have a
20, d= 1%, and the two values
60
of x, 60, and 3600; hence (S) =
I 9
26
.I
× 20
98
1
38157212, the fpace paffed
Over
188
THE SUMMATION
3600
over in one minute; and (S) =
2
188
X 20 399'99999999999999999744
73948+, the ſpace deſcribed in one hour. Alſo (x) =
20
1-18
= 400 miles, the
fum ad infinitum.
3.
Required the fum of z (10) terms of the feries
2
2
5+3)² + 5+01² + 5+91² + &c. Z².
Comparing this feries with N° 4. we get m = 5, n = 3, r = 2, (≈ = 10),
hence Z (m+zn) = 35; and therefore
9
3
2
(S)
5+3%
5+32 5+3%
5³
52
5
+
+
= (in the cafe pro-
2
2
9
2
2
35
8
9
2
2
9
35
3
pofed) +
35
2
+
•28 = 5635.
Required the fum of 5 terms of the feries
г² +
94
1 2
플
​befe
2· в + 3 • г² + 4.1 + &c. Z¹.
I
+3
This feries as it now ftands does not agree with any of the forms in this order, but
by taking away the firft term it is reducible to
I
12 1/4
+ в +
14
2
+ 3 ²² + +
1, r = 1, (≈ = 4), and
which coincides with N° 4.
Z
z ( = 1 + 1 ) = ↓
4
8.773
3
8.1 t
3
+
Hence m
+ &c.
= 1/1/1 , n = 1,
Thefe values fubftituted in the formula (S) produce
जिनेत
1/2
विगत
2
Ha
플​.
1.1
I
+
3.4.
1.1
3.4.1
2
3
플
​3
•
•
4
4
+ &c.
52
2 · 3 · 4 · 5.6. //
+ &c. = 3.6912 &c. to which adding
(the firft term of the feries) we have 4'1912 &c. the fum of 5 terms extremely
near, which would have ftill been more exact had more terms been ufed; but can
never be accurate becauſe n is here a fraction.
But the fummation of theſe kinds of feries may be often more elegantly effected, by
changing the general feries into a fimilar form with the propofed one. So in the cafe
propofed, if we add 1 to each of the formulæ (F) and (S), make m and ʼn each equal
to 1, then multiply the whole by n", the refult will be (F) n'+2n + 3n+ &c.nz
Z
Z
rz
พ
(5) n' x :
+ +
r+I
2
3.4
I
I
&c.
+
r+i 2 3.4
z retains its proper value, viz. the number of terms to be fummed.
+
==
+ &c. where
Required
OF SERIES.
189
Required the number of cannon-ſhot in a fquare pile, the fide of which is 50.
The feries will be 1 + 2² + 3² + 4² + &c. 50; the fum of which by N°
3
50
3
50
+ +
2
50
6
= 42925, expreffing the number of ſhot in ſuch a pile.
7. is
Required the number of folid inches in a pyramid composed of 1000 ftones of a cubical
figure, the length of the fide of the highest stone being one inch, of the fecond two inches,
of the third three inches, &c.
3
The feries will evidently he 1 + 2³ + 3³ + 4³ + &c. 1000; and by N° 8. the
4
=250500250000, the folid inches required.
•3
1000
fum will be
+
4
1000
2
+
1000
4
4.
Theſe figurate numbers are of great ufe in Play, Combinations, Railing the Powers
of Binomials, Refiduals, Apotomes, &c. They always refer to, or reprefent ſome
geometrical figure, and may be either linear, fuperficial, or folid. For the various
kinds of figurate numbers, fee Ozanam's Didionaire Mathematique, p. 29, &c. or
Paſcal's De Triangulo Arithmetico, and his Combinationes. The figurate numbers here
made uſe of after the 3d order, which is the pyramido triangular, are thoſe termed
pyramido-pyramidal, which are denominated in order, triangular, quadrangular,
pentangular, &c.
Required the number of combinations of m in n things.
The fum of n ከ-
required.
I terms in order m is the number of combinations
How often can a different fet at whift be made by 10 persons?
By comparing the terms we have m = 4, n = 10; whence n m-1=7,
therefore the ſum of 7 (2) terms in order 4th, will be 7.8
number of ſets as required.
7.8 9.10
1.2.3.4
: 7, and
210, the
Required the number of the coefficients in each term of the product of n factors, x+b,
x+c, &c.
The coefficient of the firft term is unity; and the number of combinations for the
coefficient of the ſecond term is 1 in z things, for that of the third 2 in ʼn things, for
that of the fourth 3 in n things, &c. Hence will the number of the coefficients of
the different powers of x be
I
190
THE SUMMATION
i.
= 1.
2.
The fum of n terms in order 1. =n.
n-
. n
3.
The fum of n- terms in order 2. =
I
I. 2
ท
4. The fum of n-2 terms in order 3.
i.n.nti
I.2.3
&c.
From hence is eaſily derived the theorem for raifing a binomial to any given power,
by fuppofing b, c, d, &c. ultimately equal to each other.
5.
Required the number of shot in a complete oblong pile conſiſting of 15 tires, the number of
Shot in the appermost tire being 32.
The uppermost tire in an oblong pile muft evidently be a fingle row; hence the
number of ſhot in each fucceffive tire will be expreſſed by the terms of the feries
32 +66 + 102 + 140 + &c. to 15 terins, Compare this feries with N° 27. and
it will appear that m = 31, e = 1, p = 0, and ≈ = 15; therefore (S) =
16. 3!
6
15 × 8.31 +
I
z
= 4960, the whole number of ſhot in ſuch a pile.
In like manner may the number of ſhot be found when the pile is a broken one,
that is, when one or more of the upper tires are wanting.
What is the fum of Z (40) terms of the feries
I
1.2 +3.4 +5.6+7.8+ &c.?
Compare this with N° 34. and we have z = 2, Z = 40; whence
= (s)
Z.Z+J.Z+2
3
=(in this caſe) 22960.
What is the expectation of a ſingle life of a given age?
Let m be the complement of life, then will m terms of N° 36. give the expectation
required. That is (S) = 1
m
m
× m =
2. Thus, fuppofing the age 50, and 86
2
2
(according to De Moivre) the extremity of old age, we have m = 36, and therefore
(S) 18 years, the time a perfon of 50 may juftly expect to continue in being.
What is the expectation of two joint lives whofe ages are A (30) and B (22) ?
Let n be the complement of the elder life A, and m that of the younger B; then
will n (56) terins of N° 37. fhew the required expectation, or that time they may
juſtly expect to continue in being together. That is (S) =
n
n
= 20 years
2
6 m
nearly.
OF
191
SERIES.
nearly. In like manner may the expectation of three joint lives be determined by
Nº 38. n, m, and t, denoting the complements of the eldeft, fecond, and youngest
lives, refpectively.
Required the fum of z (6) terms of the feries
60. 100 + 55.97 + 40.94 + &c.
3;
8=3;
This correfponds to N° 40. where we find a 60, d = 5, b = 100,
hence (S) 6000 z-340. X z z + { × 2. 2−1.2%−1, = (when z = 6}
26625.
What is the fum of Z (10) terms of the feries
15 +40 + 77 +126+ &c.?
I,
This feries is adapted to N° 42. the affumed arith. prog. being 2, 5, 8, 11, &c.
where x = 3, $ A:
= 2, a = 2, d = 3; and thence A = 15, B = 40, C = 77,
D= 126, &c. Theſe values being written in the formula produce
(S) = 15 Z + 25 X
2715.
Z.Z-I
2
+ 2 × Z. Z−1. Z−2 = (when Z = 10)
=
If the affumed progreffion be 3, 5, 7, 9, &c. and x = 4, Z = 7, the fum
will be found = 952.
Required the fum of the whole feries
6.
3
38 × 77 × 1² + 35 × 74 × √ + 34 X 71 X + &c.
alfo the fum of any number of the first terms thereof.
This feries belongs to N° 43. where by compariſon we get b = 40, m = 2,
c = 80, n = 3, c = 1, p = 2, 93; hence by ſubſtitution (2)
e =
129
= 825
343
the ſum of the whole feries; and, fuppofing z = 4, (S) = 825
129
1153
2
343
1372
15
822
>
the fum of 4 initial terms.
28
Required the whole fum, alfo the fum of any number of initial terms of the feries
12 × 20 X 16 ×
10 × 1
+ 10 × 18 × 14 ×
8 ×
8 × 31³
+
8 X 16 X 12 X
6 X
+
6 × 14 × 10 X
4 X
&c. &c.
&c. &c. &c. &c.
=
This feries belongs to N° 44. where we have a 14, b = 22, c = 18, d = 12,
*
m = 2, p = 1, 92; from whence A (abcd) = 66528, B (abs + a b d + a c d
E e
102
SUMMATION
THE
( c
hed) 17016, C (ac+ad+ab+bc+bd+cd) = 1604, D (a+b+c+d) =
66, E. &c. = 0. Thefe values being fubftituted in the formula (E) produce
99.4*20800; 8125, the fum of the whole feries. Now to find the fum of z terms,
c
ſuppoſe 3, we have a (a−zm) = 8, b (b−xm) = 16, ć (c−zm) = 12,
á (d—zm) = 6; hence Á (abcd) = 9216, B
= 9216, В (b c d + acd+abd+abc) = 3648,
Ć (á ć+ c d + &c.) = 632, D (á+b+ítá) = 42, and p (p+zq) = 7. Theſe
values being written the formula (S) the refult is 9924.2080078125—2080078125.
9924, the ſum of three initial terms of the propoſed ſeries.
Orders 7 and 8 are fufficiently exemplified by the different numerical values of n.
9.
Required the whole fum, and the fum of ≈ (5) terms of the feries
Τ
I
I
I
I
+
+
+
+
+ &c.
100
90
80
70
60
By comparing this feries with Nº 66. we find a = 100, d = 10, and p = 2;
theſe values ſubſtituted in the formuła (S) produce '000861566, the ſum of 5 initial
terms. And it is evident that the feries terminates in 10 terms; therefore by writing
10 for z in the fame formula, we have the fum of the whole ſeries
I nearly.
154
co6497677 =
Nos 71, 72, 73, are elucidated in prop. 14. of the preceding part, and in
P. 122, 123.
Required the fum of the infinite feries
10.
I
I
I
+
+
+ &c.
2.3.4.5 3.4.5.6 4.5.6.7
This correfponds with N° 78. where n = 2, and therefore () =
I
11253
72
What is the fum of the infinite feries
II.
I
2
3 · 4
I
I
I
+
+
1.3 3.7 5. II
+ &c.?
I
The general term of the feries will evidently be
;
whence by com-
2% −1.4%— I
fraction
pariſon we find m = p = − 1, 9 = 2, n = 4, and therefore, which being a
OF SERIES.
193
fraction the fum is expreffed by 2. (E)
fluent put *
I
✰
X
хх
To obtain the
I
=y, then is x y², x = y², and 4yj; hence the expreffion
y
y
becomes 21-212-2f. The fluent of
S:
yo. 1-y
y².
-S:
3. 1
=
I
the first term is had by Emerfon's flux. form 6th, and that of the fecond by form 8th,
cafe 4th. Theſe fluents being connected by their proper figns, and reduced, the
value of y restored, and lafily x made 1, the reſult is circ. arc. rad. 1, fin. ✔,
— { L. 2. = 43882475, the fum of the propofed infinite feries; which agrees
with that given by the Rev. Mr. Wildhore in Mr. Hutton's Math. Miſc. p. 286.
For other examples to this order fee p. 17 to 26.
12.
I
I
I
I
I
Required the fum of the infinite ferics
+
+
2.2
3.4 4.8
5. 16
6.32
I
+ &c.
7.64
I
The general term of the feries is
; hence by comparing the terms we ob-
ช
I-
2
tain p = q = 1, and 6 = 2, which ſubſtituted in the formula (2) produces
x x
2+x
and by reduction
S-25
x
2+
2 L.2+x. Now make x
Now make x = o, and the
reſult is 2 L. 2; hence the correct Aluent is x 2 L. 2 + x + 2 L. 2
(when 1) 1 + 2 L. 2 - 2 L. 3 =
x =
infinitum.
-
1890698, the fum of the feries ad
It may here be obferved that the fummation in this order of feries is no way facili-
tated by the formula (S) whatever the figns of the propofed feries are, nor by the
formula () when the figns are affirmative, the fluent of the refulting expreffion in
each cafe not being attainable but by throwing it into a feries, the terms of which ul-
timately become the fame as thofe of the propofed one. See another example,
P. 23. 35.
For examples to the orders 13, 14, 15, 16, 17, 18, and 19, fee p. 44, 65, 73,
69, 83, 98, and 109, refpectively.
20.
Required the fum of the infinite feries 2 ≈ + z² + { ≈³ + ‡ ≈+ + } z³ + &c.
Here A2, B, C, &c. Take az, 82, y = 0, &= 0,
&c. then will
d
E e 2
*. 23
194
THE SUMMATION
1
ВА
Hence () =
a + B
α
2x + Bx²
ΟΙ
a Z²
+ B.z
or
a. 1 z ³² + B. 1 z+
= C.
&c.
&c.
42
the fum required.
2-
Whoever would fee more of the nature of thefe feries, may confult De Moivre's
Mifcell. Analyt. Lib. II. cap. II. and Lib. IV. cap. I. or Simpſon's Eſſays, p. 96.
21.
To exemplify this approximating theorem, let the length of the circular arc, when the radius
is 1, and tangent t, be required.
hence by compariſon we have a = 1,
n = 2; and by fubftitution and reducing we have
£3.
ts
t
The feries for the length of the arc is well known to be t
+
+ &c.
3
5
7
I, @
I
f =
d =
39
†, &c. r = 1,
t 44
(x) =
4
325
बड़े है + और है 3
बैंड + 18s t + इंटेड
=
1, it becomes 785 very
an approximate value for the length of the arch. If t
nearly; which, in this extreme cafe, is the fum of near 500 initial terms of the pro-
pofed feries.
22.
Required the fum of the infinite feries 1 + 2 + 3*² + 4x³ + &c. where x is lefs
than I.
This feries coincides with N° 9o. where, by comparing the terms, we find a = 1,
3 = 2, y = 3, &c. Ú
I
(x) =
I X
+
**
= 1,
Ď
o, &c. hence (taking x negative in this cafe)
I
Required the area of the right-angled hyperbola, the principal diameter being 1, and the
abcifs x.
The feries expreffing this area is
3
5
៖
4*
3
2 x
2 x
2.3 ༢ཚ
+
+
+
5
4.7 4.0.9
2.3.5
4.6.8.11
+ &c.
But in order that
this feries may converge more quickly, let a few of the initial
terms be fummed, fuppofe 4; then will the coefficients of the remaining terms be
2.3.5
B =
4.
6.8.11
2.3.5.7
4.6.8.10.13.
&c. or a = '0142, 00841, y =
B
•00546,
Ꮧ
f
OF SERIES.
195
*00546, &c. and thence Ď = −00579, Ď = +·00284, Ü=00156. Now
affume any value for x, fuppofe 1, then is (E) = oogo, to which add the fum of
the first four terms 1.6896, and the fum of the whole feries is 16806, &c. If x = {},
the reſult is 536787.
Required the fum of the feries
This feries is refolvable into
-플
​But c + 20 x + 3/3/3 c
-3
2
23.
મ
X
+
3
x
40
8c
5
+ &c.
I
+
C
-플
​2
Min
C
20
+
I
2
+ &c.
divided by numbers in arith
x
+
3
c-Z
x²+ &c. appears to be
ed
플
​,
and the terms are
prog. the propofed feries therefore correfponds to N° 93.
Hence by comparing the terms we have a = c, z = 1, r = 1; and by ſubſtitu-
tion, the fum of the whole feries is =
C - X
HIN
HIN
플
​, or
C
HIN
c-x
४
મ
Required the fum of the infinite feries
24.
a
6 a x
+
+
I
3
15 at x²
6
20a³ ³
15a² 4
+
+
+ &c.
IO
15
This feries appears to be the 6th power of the binomial a+, each term being di-
vided by the correſponding value of the fifth order of figurate numbers. Here then by
compariſon we have z = 6, r+15, orr = 4; and by fubftitution
a + *
(x) =
3
a² + 10 ~° x + 45 a˚ x² + 180 a7 x³
7 · 4 · 3 · 4. *•*
N. B. In this and the preceding order, add to the numerator of the general formula
(E) the words minus r initial terms.
25.
Required the fum of the infinite feries 4x
+
2
16x3
3
25 *
+ &c.
4
{{ x² + 1 ׳ −
x²
*་
This feries is compounded of the infinite ferics x
+ &c. and
of the feries of fquare numbers 2. 2, 3. 3, 4. 4, &c. the former of which is well
known to express the hyp. log. of 1; hence we have NL.1+x, and Ñ =
Ń
Therefore the propofed feries coincides with N 98, from whence we have
i
1 + x
m = 2, p = 2, r = 1, ♪ = 1, and R (
(=
= 1-X N +
|
x²)
=
= N +
R =
Ŕ
196
THE SUMMATION
2 x
-1
x x
; and confequently (2) = (p=r × R +
Ŕ
I + *
I + a
= L.i+*+
3 **
2
I + x
I + x)
X
Ꭱ
=) B+
2 x
1 + x
26.
1 3x² is
That is the fum of the infinite feries x3 + x + 15×¹³ + 21*²¹ + &c.?
I
I
A'
x
By comparing the terms we have m = 3, n = 6; hence (2) = S**
L.
1 + x
I
3
ぽ
​If x, the fum is L. = '0418856, &c.
7
I
27.
2
Required the fum of the infinite feries
4z2
624
+
&c.
3.5 5.7 7.9
ʼn
This feries correfponds with N° 101; from whence we get = 2, a = 2, p = 3,
9 = 5, A = − 1, B, C = o, &c. Now, for m in the formula (E) N° 100,
affume any whole number at pleaſure, fuppofe 1, then will R in this cafe be=
2
ż
+z
N° 101, we have
circ. arc. rad. 1, tang. Z.
3+~2
225
And by fubftitution in the formula (E)
X circ. arc. rad. 1, tang. z
3
224
the fum of the pro-
pofed feries.
Required the fum of the infinite feries
x
2.3
+
.2.4
+
•
I 3·4
3.4.5 4.5.9
+
5.6.14
+ &c.
Divide each term by 2.v, and the feries becomes
I
+
•
I 2
•
2
3.4
2
+
•
3 · 4 · 5
3. 4. 5. 6
+ &c. which agrees with N° 103; whence we have n = 1, p = 1,
v+ 1 = 4, or v = 3; and affuming for m (in N° 100.) any whole number, fup-
pofe 1, the value of R is found to be
1. Hence by fubftitution,
S
x
I
L.
I-X
I
X
I I
5
I
I
18
6x
3x²
multiplying by 2, and reducing, there arifes
fum required.
+
3x3
>
the
Required
OF
197
SERIES.
Required the fum of the infinite feries
•
I
I .2 3.4.5
I
or
42
グ
​+
+
+ &c.
4. 5. 6. 7. 8
7.8.9.10.II
4
7
+
+
+ &c.
I I
2.3.4.5 5.6.7.8 8.9.10.
This feries in either forin correfponds with N 104.
we have a = 1, b =
1, b = 1, p = 1, q = 2,
B, C = 1, D=
Comparing the firft form,
4, t = 5, "
12 = 3; alfo
3, E. Hence qp, r- p = 2, s− p = 3,
t-p = 4; and the formula (E) becomes
quired fum.
209
3
==
the re-
210 280'
2, r = 3,
-
I
5
117
+
6
4
56
28.
4x3
6xs
8 x?
+
+
+
+ &c. ?
What is the fum of the infinite feries
2 λ
"
1. 3. 5 3.5.7 5.7.9 7.9.11
Comparing this feries with N 105. we find a = 2, m = 2, p = 1, 9 = 3;
I
r = 5; alſo A = ¹, B = 1, C =
3
3 D = o, &c. P = x + + + &c.
3
5
[ + *
P
p
= L.
e=
R =
P-- Hence (x) = +2x-3xL.
**
*
X
X
+x
3
+
If we take x = 1, we have the fum of the infinite feries
I
8*
2
4
I
+
+ &c. =
; which agrees with formula 1. (E) in order 17.
1
3.5
3 · 5 · Y
4
29.
IX 10 y²
Let the feries propoſed be 1
+
2 X 4
1. 3 X 10.13 µ+
2.4X4.7
1.3.5×10.13. 16y6
2.4.6X4.7.10
+
1 . 3 · 5 · 7 × 10. 13. 16. 19 y
2.4.6.8 × 4 • 7 ∙ 10 . 13
&c.
This feries coincides with N`III. where, by compariſon, we find n = 10, r = 4,
m = 3, p = 1, v = 2, x = y', and q (nr) = 6. Subftituting theie values in
the formula (E), and reducing, we fi›d
XI-
37
4.1+ y`
I
27 y
+
>
I 12
I + y²
for the required value of the feries.
N. B. In this N°. it fhould have Teen
I
(E)
•
p
X
X
+
.ย
9
p.pt.
r. r + m. บ 20. IF N
•
9 • ?
•
I
9-2 in. p p+v.p+27
&c.
3
20.30. 17 &
r + 1.r+2 m. v.2 v
+1
30
198
THE SUMMATION
30.
Let the fum of this infinite feries be required
b
a +
3x5_b-I
IX6
a
z +
3.4X5-7 6-2
z² +
a
1.2X6.8
Ι
3 · 4 · 5 × 5·7·9 6-3 z³ + &c.
2.3X6.8.10
a
Z
b
3X5
This feries by reduction, and writing ≈ for
becomes a X: 1 +
x +
a
IX6
3.4 × 5.7
•
I 2 X 6.8
x²+ &c. which coincides with N° 113. Whence n = 3, m=1, p = 5,
8.s+m
w = 2, r = 1, v = 6, 9 = 2,
q 2, 5- ·I, t = 1, k = 1, and h =
; and
6
by fubftitution all the terms vaniſh, except that where s+m is found in the denomi-
I
nator, which then becomes
Confequently multiplying by a', and reſtoring
I
b
a
the value of x, we have () =
which was required.
༤
I
a
The orders 31, 32, 33, 34, 35, need no particular illuftration.
36.
Required the fum of ≈ terms of the feries 1 . 2² + 2 . 3² + 3 · 4² + 4 • 5² + 5 . 6³
+ &c.
2
I
2
The general term of this feries will evidently be z.z+1, and therefore the
2+1th term, +1.+2. Put z+2=x, then is the general term of the feries
expreffed by x-1.x², or x³-x²; and comparing this with the general feries, we
have a 1, B =
Subftitute thefe values in
1, y = 0, &c. ≈≈ 3, and ▲ = 1.
z
the formula (S) and there arifes - ³ + 2 x² - 1 × - To. Now, it is ob-
3/8 x
3
4
4
X
2
I
vious, that when the expreffion for the fummatrix is ſuppoſed to be o,
alſo be = 0; and confeq. from the above affumption we have x = 2.
of x being written in the above expreffion, the refult is
been ; this term being therefore eliminated we have
✩✩
that z muſt
This value
rio, which ſhould have
20
.3
- x3 + x² - ↓ », or
5/8 32
2+2
4
3
2
5.2+2
6
+
3.2+2
4
x + 2
6
,
the correct fummatrix for z initial
terms of the propofed feries.
Let the fum of z terms of the feries 1 . 2° • 3³ + 3 · 4² · 5³ + 5 · 6² . 7³ + &c. be
required.
2
The general or zih term of the feries will be 2%-1.2% 2x+1³, and the
z+ith term 2.Z+I−1.2. ≈ + 1′
3
2
3
2. %+1+1 or 2z+1.2Z +2
•
2≈+3' •
Now,
O F
199
SERIE S.
4
Now, in order to make this correſpond with the form, put the root of the laſt factor
(2x+3)=x, then will the z+1th term be x-2.x-1)². *³, or x6 — 4*5 + 5**
I x²,
+5x³-
2x³. This expreffion being compared with the general form, we have a 1,
B = 4, 7 = 5,
5, d = −2, e, &c. = 0, % = 6, and A = 1; which values being
ſubſtituted in the formula (S), and the refult corrected as in the laft example, there
α =
7
6
x
arifes
+
5x 7x5 734
14
6
+ 3*³ +
5x2 4x
7
3 2x+3
or
2
12
3
7
4
2. 7
5.2x+3
6
S
4
3
7.2x+3
+
73·2%+31
3·2x+3
5·2%+3
+
+
2
3
2. I
3.4
I
3
4.2≈+3
7
3
2, the fummatrix for z initial terms; which agrees with that given by
Mr. Ainſworth in the Ladies and Gentlemen's Diary, 1779, P. 32, for the ſame
feries.
37.
Required the fum of x-1 terms of the feries
2
4
4
7
16 32
8
+ + + + + &c.
IO 13 16
By comparing the terms we find r = 2, n = 1, d= 3; hence
=
2≈
6
(S) =
2
12
6
༤
27
2
+
X
+
X
X
+ &c.
1+32
1+3.≈+1
1+ 32
1+3.2+2 +3.2+1
1+32
2
6
2
12 6
2
X
X
&c.
4
7
4
10 7
4
When is less than 1, this theorem coincides with order 12; from whence the
fummation is more elegantly brought out in fuch cafe.
Required the fum of z terms of the feries
38.
2
22
23
2
mp
CA
+
I. 5
2.6
+
+
+ &c.
3.7
4.8
The terms being compared, we have m2, n = 4; and fumming up 5 terms of
the feries, Q = 2*32539680, v = 6.
I
Z
+
quired fum.
I
+ &c.
≈.2+1.≈+ 2
Hence (S) = 2.3253 &c. + 2 x :
X:
I
I
25 x : +
42 330
I
+
+ &c. the re-
3024
Ff
39.
200
THE SUMMATION
1
39.
Required the fum of the infinite feries
2 x
*9
2.1.2 x'I
+
+
9
9. II
2.1.2.3 x¹³
9.11.13
13
+ &c.
I
27 x
+
JI
2
This feries is refolvable into
x² X : I +
N° 162. we get m
"
A
27
A
23
= }, Ä = 3.
2
X
+
= 1, n = 2, 9
1, r = 1, s = 1, a
Hence by fubftitution in the formula (E) there arifes
1. 2 • 3x5
I L
2
1 I
I3
+ &c. which being compared with
༡- 3, t =
½, Á = 1,
I - Y
-7X
5/2
2
X
24
I
x +
7 22:3
3
23x5
= 7x-49׳+
I 5
x³-7
15
x
2
I - X
x
I-
Now take any number lefs than 1, fuppofe, and the laft expreffion becomes
x
3′5 − 2·0416 +335416 - 3′40992 × 523598 =
3*40992 × *523598 = .008333 &c. the fum of the
propofed infinite feries in this cafe;
being the flux. of the circ. arc. rad. 1,
fine x (1), arc. of 30° =
6.2831853
12
='523598 &c.
40.
Let the feries propoſed be
4
x
x5
46
+
&c.
1.2.3.4 2.3.4.5
3.4.5.6
This feries agrees with N° 167. from whence we have 3; and by writing
this value in the formula (E), there arifes
6
2
3
× 1+ × L. 1+x-x-x²-11 x³. If x = 1, the fum of the feries is
1 X 8 X 0·6931472-1-2-03530738.
Required the fum of the infinite feries
I
22
+
3.4 4.5 6
•
41.
+
+
•
I 2
24
1. 2. 3. 2º
7. 8 9.10
+ &c.
6.
5.6.7.8
Compare this with the form N° 169. and it will appear that = 1, n = 2; which
laft value being an even pofitive number, and the figns of the propofed feries affirma-
tive, indicate that the fum is exhibited by the formula 2. (E). Now when x is 1,
the circ. arc. is a quadrant, and therefore S, the fine of (n times) twice that arc.
1
is o; confeq. the refult will be 2. (E) = х
n
+2
24
If
O F
201
SERIE S.
I
I
If the feries be
+
2.3
I
I
will become
+
+
3.4.5 4.5.6.7
+ &c. which is reducible to
I.2
+ &c. divide it by 8, and it
2.3.8
3.4.5.8
4
5
1
I
1.22.
लाल
X :
+
+
2.3
3.4.5
1 . 2 . 2.
4.5.6.7
+
1. 2. 3. 26
5.6.7.8.9
I
•
+ &c.
Now compare this feries with the form, and it will appear that n = 1, x, and
therefore C = √. Hence the correſponding formula is 4. (E), n being an odd po-
ſitive number, and the figns of the propofed feries +; and the fum of the latter feries
will be expreffed by X 7th the circumference of the circle, rad. 1, which
multiplied by 8 becomes 2-3X th the circumference of the circle, rad. 1, =
•18611 &c. the fum of the propoſed ſeries.
4
✔
42.
Required the fum of the feries
8
• 3
20.12.15 (≈) + 10 X 20.10.13 (2-1) +10. X 20.12. 3 (≈−2) + 10
• 2.3 x 20. &c. to 10. %. (z).
By comparing the terms, we find n = 20, m
? ?
= 10; hence
30
(S) = X X
29 28
X &c. to ≈ terms.
If z
If ≈
ལ =
5, the fum of the whole feries.
I
2
3
alternately, in the
will be 139506.
N. B. The terms of this feries are erroneously given + and
Meditationes Analyticæ; they ſhould be all poſitive.
What is the value of z terms of the feries
2
43.
2
2
27 x 44 x 66 x 85 x 9° x Sic.
This feries is reducible to 2. L. 2 + 4². L. 4 + 6. L.6 + &c. which being
compared with N° 173. we get a = 1, d= 1, p = 2; and by writing thefe values
in the formula (S) there arifes
I
I
ΤΣ
1 × 1+2≈ × L.1+2≈−÷X1+22× L.1+2≈−18×1+23² + 15-15×1+28≈+is,
the required fum.
44.
As the truth of the theorems given under this order has been much controverted by
feveral mathematicians of refpectable abilities, we fhall therefore endeavour to explain
the conſtruction of the fundamental formula, as alfo the law of continuation which
the ſeries obferves, as laid down by the learned author, before we give any
examples.
* Dr. Waring.
Ff2
It
202
THE SUMMATION
It is demonftrated in Newton's Univ. Arith. p. 251. that if a denotes the fum of
MI
#
2
+9x
the roots of an adfected equation pa
&c. b the ſum of the
fquares of the roots, c the fum of the cubes, d the fun of the biquadrates, e the fum
of the furfolids, &c. then will pa, p² - 2 q = b, pb-qa+ 3r = c,
pc - qb + ra - 4sd, &c. Hence, by obferving the law of the fucceffive ex-
preffions, and continually fubftituting the preceding one, we have
Ilp = a,
2 pa-29
b,
3pb-9a+3r = c,
p.
= p-29.
p² - 39p+3r.
4 pc-gb+ra- 4s = dy
4
= p² - 4 9 p² + 4"p
-45
+29`
5 pd―qc+rb-sa+5t = e,= p² - 5 9 p² + 5 r p²
rp²
-52 +5+
+59² S² -59r'
6 pe-qd+re-sb+ta-bv = ƒ,|= pˆ − 6 q p¹ + 6 r p³
-
-65
p²
+99²
+6t
129 r
7\pf-qe+rd-sc+tb¬va+7w,|=p'−7985 +7rp+
+149°
03
P
-60
+695
} +
-70
29
3
+3x²
+7w
+71
+1495 -7qt
P
-21gr
-793
+79²r°
+7m²
-7rs
&c.
&c.
From whence plainly appear the conftruction and the relation of the terms of the
formula, as given in the Mifcell. Analyt. cap. 1. lem. 1. and the law of
the literal products or coefficients of the feveral powers of p will from hence be
alfo very obvious. For fince the abfolute number or laft term of any ad-
fected equation is effentially of the fame dimenfion as the greateſt power of the un-
known quantity, being the product of all the roots, the literal quantities p, q, r, s,
t, &c. are here faid to be refpectively of 1, 2, 3, 4, 5, &c. dimenfions; hence, in
forming the literal coefficients for any given term of the feries, as many combina-
tions or products of the preceding letters muſt be taken as are found to produce the
fame dimenfion as the propoſed term is diſtant from the firft. So in finding the literal
coefficients for the 7th term, all the poffible combinations which can be formed of the
preceding letters of which the dimenfions arife to that number, are w, q t, q³ r, r s,
where the fums of the indices denoting their dimenfions ar cach 7.
The law of the unciæ or numeral coefficients to each of the literal quantities or
products is this; Suppofe q &c. p to be fome term of the feries, where µ, », &c.
denote the proper power of the letter, and not the dim lion of it according to its
γ
$21-14
=
7 2 5 4 3 3 4
ولا
place
OF
203
SERIES.
place in the feries, then will the uncia to that particular literal product or coefficient
of p, be expreſſed by
m. m u+1.m−u+2.m−u+3. &c. to µ + » + &c. terms
I 2 3 (μ) X I 2.3 () x &c.
•
•
Thus, let the unciæ of each of the literal coefficients of the 7th term of the feries be
required, and we ſhall have m for the uncia of w, becauſe the proper power of w
is I;
m m- ·u + I
I X I
μ
7—6 for that of qt, becauſe µ + › = 1 + 1 = 2, and u
=m.m —
denotes the distance of the propofed term counted from the firft;
=m.m m-5
ท -6
2
m.m-u+1.m-u + I
I. 2 X I
m
•
m
น + I
1 X I
for that of q`r, µ being =2, and = 1; and
m.m-6 for that of r s. In like manner are all the other terms formed, as in
the formula (S) N° 174. which exactly agree with thofe arifing from the method of
fubftitution. Hence we fee with how much truth the author of Obfervations on the
Mifcell. Analytica has afferted" that if the feries be formed according to the law here
laid down, it will be wrong in every ftep after the third." But this author, as well
as the reft that have objected to this curious theorem, muft, it is evident, have been
unacquainted with its inveftigation, and alfo with the connexion and application of
its terms. We ſhall now illuftrate it by a few examples.
Let the equation first propoſed be x+3x-10= 0, the roots of which are 2 and - 5.
Comparing this equation with the general one in the formula, we find p =
-39
Hence when m2, we have p2-299+20 = 29,
9=
10, r, s, &c. = o.
the fum of the fquares of the roots. If m = 3, p³-39 p = −27—90 = −117,
is the fum of the cubes. If m = 4, p² - 4 q p² + 2 q
If m = 4, p*— 4 9 p² + 2 q² = 81 +360+200 = 641, is
the fum of the biquadrates. If m = 5, p² - 59 p² + 5 9 p = -243-1350-1500
=-3093, the fun of the furfolids. If m = 6, pˆ — 6 q p + + 9 q` p² - 2 q² =
729 + 4860 + 8100+ 2000 = 15689, the fum of the fquare-cubes. Et fic in
reliquis.
Let the propofed equation be x³ +7.x²+2x−40 = 0, of which the roots are 2,
− 5, —4.
Here we have = −7, 9 = 2, r = 40, s, &c. = o. If m = 4, p³ − + 9 p² +
p
4rp+29² = 2401-392-1120+8 = 897, is the fum of the fourth power of the
roots.
Let the given equation be x+3x† − 2 3x² → 27 ✰²+166 x−120 = 0, the roots being
I, 3, 2, -4, -5°
-
Here p =
3,923, r =
then is p-59 p³ +5 rp² -55
s
27, ♪ = 166, t = 120, 9, &c. . = 0. If % -5,
+517 = 243 - 3105 + 1215 + 24
I
+51.
-591
7935
204
THE SUMM
SUMMATION
7935+600+31053873, the fum of the fifth power of the roots. If m = 6,
+61 +695) =729+11178-4374 + 33885
p' − 6 7 p² + 6 r p³ -- 6s 7
3
+9q² } ¹² — 12 gr } P — 29³ (1
+3x²
−24516+3613 = 20515, the ſum of the fquarc-cubes.
8
I
If the equation be a-x+x+= o, we have p, 9 = 1, r = -};
and when = 2, p²−29 = £;; = 1, is the fum of the fquares of the roots. If
m = 3, p³ - 39p+3 r = 125 — }{ == 174, is the fum of the cubes.
3
I
64
7 I
64
From hence is eafily deduced that elegant method of folving equations, where the
fum, ſum of the fquares, fum of the cubes, &c. of any number of quantities are
known, which the late ingenious Mr. Coughron has given in Profeffor Hutton's
Mathematical Mifcellany, P. 124•
Let the given equation be x³-3x² + x²+3×− 2 = 0,
to find the fum a+b+c+d.
Comparing this with N 175. we find p = 3, 9 = 1, r = − 3, s = -4, and
2 = 4. Hence
(S) = p-p
2-p-5p³ + 4p+
3
2
<
3 − 2 p − 5 p² + 4 p³ _ s x 4 - 8 p + 4 p²
+
92 X
4- 12 p + 12 p² - 4 p³
2.1-p
= 120-47-45+8+2=38.
I
Let the equation propoſed be x³-3x² + 13×−} = 0, to find the fum a+b+c+d+ &c.
ad infinitum.
3
Herep, 91, r = }; hence (E)
p-29+3r
I-p+q-r Ι 출 ​+
I 3
+ 풀
​3
I 3
18 す
​=35.
The truth of which conclufion appears from finding the feveral fums of the feries of
powers, formed from the roots of the given equation. So the roots of the above
equation are found to be,, and; hence by N° 122. we have the fums of the
infinite feries+} + {; + &c. = }, {+} +}+&c. = 1, } +$+37+&c. = 2,
and the fum of the three feries 3, as before.
I
=
I
4
Let the equation propoſed be x*- x³ + 1x-16 = 0, to find the fum a+b+c+d+ &c.
in infinitum.
ป
In this example p = 1, 90, TS-
(E) =
p+ 3r-45
I-prits
found to be 1,,,
I
- + D +
3
I 4
I
r
; hence
= 2. Now the roots of this equation are
1-2 +1
1-1+1-16
4
-; hence 3 × : 1+1++ &c. 3, and by N° 121.
a
1³ + &c. or ~ 1+1-} + ↓ ~ &c. ≈ −; confequently the
I
사
​J
fun of the powers of the roots continued ad infinitum is 323, as before.
Let
}
OF
205
SERIES.
Let the quadratic equation x² + 3x-10=0, be propofed, it is required to find the fum
of the product of each root into the fquare of the other.
The fum of the roots of this equation is 3, and the fum of the fquares 29, by
N° 174•
s¹+² = a¹
Hence by N° 176. we have S¹ =
a + B
=
1+2
-117, confeq. A =
Ꮪ
-117, and A-B — 30.
—
3, S² = a² + B² = 29, and
S' X S² =
2
S¹ × S² = −87, B = s¹+ =
Let the propofed equation be x³+3x+ − 23 ׳ — 27 x² + 166x-120 = 0.
By N° 174. we have
3
C
3
Sª = a+ß+r+d+e = − 3, 5º = a² + ß² + &c. = 55, S° = ∞³ +ß³ +&c. = −153,
sa+b=a³ + B² + &c. =−153, Sª+c=aª +B*+&c. =979, S°+' = a³ + ß³ +&c. =−3873,
sa+b+c=a° +B+ &c. 20515, to find the fum of
3
α
3
α
α
3
E
we find A = S x S x S′ = 25245, B = Sa+ỏ
X
aß² y³ ta B² D3 + a ß²è³ + ß æ² y³ ‡ ß æ² d³ + Ba² ε³ +ya² ß³ + &c.
By the formula Nº 176.
× Sˆ
sc +sa+c xso +so+c x sa =
153 +979 × 55-3873 × 3 = 88873, C=
sa+b+c= 20515. Therefore (S) = A−B+2C = −22598.
2
Required the value of the infinite feries
45.
4
✅ 18+ 189
4
✓ 18+189
4
√ 18+189
&c.
This correfponds to N° 179. from whence () = the root of the equation ׳ — 18×
-1890; and therefore () == 3, the fum required.
Required the value of the infinite feries
√8+64
2
:
4+20
4
√8+64
2
4+120
&c.
By N° 180. we have (E) = the root of the equation -8° × 4*+**—8′ X 120
2
64° × x, from whence x (E) is found = 2,
16.
206
THE SUMMATION
Required the fum of 10 terms of the feries
1.2.
46.
I
22
+
3 • 4 • 5 4.5.6.7.8
7.8+
32
7.8.9.10.11
+ &c.
By fubftituting fucceffively the feries of natural numbers for z in N° 186. we find
the feries arifing to coincide with the propofed one. Hence by writing 10 for z in
the formula (S) we have
propofed feries.
ΙΟ I I
12.31. 32
•
55
the fum of 10 initial terms of the
2
5959
To exemplify the third remark to this order of feries, let the fum of z terms of the
2
I
feries
+
r
73
2 3° 4
+ + +
2
5²
+ &c. be inveftigated.
The zth term is
2
א
12
and the zith term
z+
x²+2x+1
g
I
Affumes Az² + B≈ + C+
D
I
+ &c. X
25
= r A x² + r B z + r C + " D
r
I
+ &c.
; now write z+1 for
Z
*
=rAx²+r
z, and the refult will be
s=rAz²+rBz+rC+
s = A ≈² + 2 Az+ A
+ D
Z
+ &c.
r = + I
I
+ B≈+ B} + &c. x
+CJ
I
r²+1; hence
ک
= + A
z²
2
+2A
+A
~rAŠ
+ B+ B
-rB + C
I
+ &c. x
I
x²+2x+1 X
≈+1
Equating the coefficients, we have A =
I
21
B =
>
C
Ι
r
= 0;
Fo; hence s
I
27
=
X z²
xx+
3
I-r
Ι r.r
་ླ
I -
r
I
r
r
2
Z
27 z
z
א
Z
39
, D, &c.
I
; which being
corrected by making = o, becomes
r+r
1
If we take r = 2, it becomes
6-6+4≈+ z² x
I
>
which
༧
2
agrees
+
- I
до
T
42
6
I
6
+
X
23
I
I
I
I
2
with the fummatrix (S) in the 2d example to
order 47.
47.
OFS E
SERIE S.
207
47.
What is the prefent worth of an annuity of a Pounds to continue z years certain, at com-
pound intereft?
Letr be the amount of 1 L. in 1 year; then will
M.
I
be the prefent worth of 1.
I
due at the end of ≈ years; hence the ſum of z terms of the ſeries
1
}
+
+
да
+&c. multiplied by a, will give the required value. Comparing as feries with
N° 190. we have n = 0, p =
I
≈; hence (E) becomes
g
I
1-1
and (S) = 1-5 ×
I
I
; confeq.
a-ap
↑ - I
I
↑ - I
is the prefent worth of the annuity.
Required the fum of z terms of the ſeries + + 8 + 16 + 3½ + &c.
Here we have n = 2, r = 2, and thence (E) = 6; alſo P = 1, Q = 2. Cen-
6+160+1600
I
feq. (S) = 6—6+4x+z² ×
z
If ≈ 40, we have (S) = 6
6—
2
= 5
549755813005
549755813888*
See T. and C. Magazine for July 1779, P. 344•
48.
Required the length of the circular arch of 30 degrees, the radius being unity.
Let r denote the radius and t the tangent, then will the arch be expreffed by
23
3r2
÷ + 5 -
+7
t
+ &c. or t x : I
++
+5
+
י7
72°
31
5ri
+ &c. or (putting
*
X
1
r√x X: 1 –
+
3
+
5
7
9
&c. Compare this with the
formula (F) and we find a = 1, b =
1/3, 6
c = }, d = 4, e, &c. and therefore
IV
Ú = ·13, Ű = •07619047619, D
•07619047619, D = 050793650793, D =
•050793650793, D = '036940836940, &c.
Now in the cafe propoſed t is =
I
,
√3
and x; from whence we have
**
I+~
X
2.3
&c. refpectively equal to
>
I
I
I
I
1 + x
I+x
4
16' 64' 256
>
&c. Thefe values
being written in the formula (continued to 7 or 8 terms) produce 90690004 &c.
which multiplied by the reſult is 5235988 &c. the length of the propofed arch,
which is true to the 6th place.
When the feries converges very flowly, the best way is to fum a few of the initial
terms by the common method, and then apply the formula to its remaining terms.
Suppofe,
G g
1
208
THE SUMMATION
Suppofe, for example, the feries -X:1
πτυ
I 1
ขบ
I. I. 3. 3
ข
X
+
X
1
S
2.2
SS
2.2.4 4
•
I. I. 3. 3· 5 · 5
4.4.6.0
2 2
•
•
X + &c. which expreffes the time of defcent of a pen-
50
dulum through the a:ch of a circle (not exceeding a quadrant) to the loweſt point;
where r denotes the radius, v the height fallen from, or verfed fine of the arch de-
fcribed, s the fine, the fourth part of the circumf. of a circle, rad. 1, and v the
time of defcent through the verfed fine v. In order to adapt this
mula, let the numeral coefficients be denoted by A, B, C, D, &c.
pofed feries become A
B v²
C&+
+
ов
X
16
S
25 C
36
25
X + &c.
ގ
feries to the for-
then will the pro-
Dvo
S
+ &c. or A
1. A
4
X
ان
+
feries to be found, and the terms next
Now fuppofe the fum of 12 initial terms of this
fucceeding are
N 7,24
0226
sit
520
+ &c. or
23° M v²+
24² 5²+
252 N v26
262 520
+
26
282 520
2720228
s
&c. or
V24
524
215 21
224
X:
232 M
252 N v2
27² Ov7
242
+
262 s²
282s+
&c. or
x
S
232 M
X
252 N
242
* +
262
272 O
282
(writing for
ບ 2
&c. which coincides with the formula;
from whence are found the values of a, b, c, &c. and their feveral differences.
Now in the cafe where the feries converges the floweft, that is, when the arch through
which the deſcent is made is exactly 90°, v is, and x 1, from whence
x
1 + x
2
29
I+x
which becauſe
I
&c. are = 1, 4, 1, &c. and by ſubſtitution (x) = '013517763515,
224
524
= 1, is the value of the remaining terms of the feries fucceeding
the 12th term. To this add 821109079506, being the fum of the 12 firft terms of
the feries (which is eafily found in the common way) and the refult is 834626843021.
TIV
•
Confeq. multiplied into this laft number will give the fum of the propofed feries;
S
that is 1.57079 &c. × r × •83462 &c. = 1°31102877 &c. which is true to the
laft figure. From whence may be inferred, that the time of deſcent of a pendulum,
or other heavy body, moving freely from a ſtate of reft by the force of gravity only
through the arch of the quadrant of a circle, is to the time of the perpend. defcent of
a body through the radius, as 13110 &c. is to 1.
This theorem will be found of great utility, as moft feries which occur in the folu-
tion of mathematical or philofophical problems are found to have their coefficients in
a decreaſing progreffion, as alfo their ſeveral orders of differences. Thoſe who are
defirous
OF
209
SERIES.
defirous to fee the demonſtration or inveftigation of this uſeful formu'a, may confult
the Philof. Tranf. Vol. LXVII. p. 187, &c. But it may juſt be obſerved to the in-
genious author *, that general formulæ obtained by induction are never fo fatisfactory
as thoſe which reſult from a pure algebraic inveſtigation.
49.
Let the propofed number be 374964, to find the sum of the numbers ariſing from all the
combinations or changes which can be made of the places of figures.
By comparing the given number with the formula, we have a, b, c, &:c. refpec-
tively to 3, 7, 4, &c. and z = 6; hence
(S) =
3+7+4+9+6+4
6
× 1.2.3 (6) × : 1+10+100 (6) = 33 × 720 × :
1+10+100 (6) = 439999560.
From hence alfo may the fum of all the changes of any progreffion of numbers be
found, by fubftituting for a+b+c+ &c. the general expreffion for the fum of that.
progreffion. Thus, in a geometrical progreffion of which the ratio is denoted by r,
the firſt (or leaft) term by a, and the number of terms by ≈, the fum of ſuch pro-
a × 1. 2. 3 (≈)
a
IX; and therefore ✈ — 1 X
r -I
21-2
greffion is expreffed by
X:
1+10+100 + (≈) is = the fum of all the changes of the numbers which compoſe
the ſeries; each term being diftinctly confidered, and eftimated according to its varia-
ble local value in the ſeries.
Let the sum of all the changes which can be made of the places of the terms of the
prog. 1, 3, 9, &c. to 6 terms, be required.
Here a = 1, r = 3, ≈ = 6; hence by fubftitution we have
(S) = 3° − 1 × 60 × : 1+10+100 (6) = 4853328450.
50.
24
Required the product of 100 terms of the feries 1, 2, 3, &c.
geom.
By comparing the terms, we get b = 20, n = 1, r = 3, ≈ 100, x = 320,
log. b = 1*3010300, log. = 2.5051500. Hence A = 1, B
I
x
13
D = 1, E = 1, F = -, &c. confeq. (S)
=
=
I
1,
I
−, C = -15,
the number of which the com.
log. is
2*505150 '4342945
*4342
+
3
3'32
18 · 32
43
932
1°30103 4342
&c.
+
3
3
3°32
°4342
*434
+
+
+ &c. or 0.408213,2°55984, the product of ico initial
18.32
terms.
9°32
* Francis Maferes, Efq.
Gg 2
51.
210
THE SUMMATION
51.
What is the fum of 25 terms of the feries
I
I
I
I
3
I'm
+ +
7
4
+
11 15+
+ &c. ?
4
I
I
This feries is reducible to
+
+
4
+ &c. or 1+ 2 4 +1+3.2
1+2
1+3.2
+ &c. which correfponds with N° 196. from whence we have m = 1, n = 2,
r = −4, ≈ = 25. Thefe values written in the formula produce
(S) = X 101)
1 2
the required fum.
3
7
− 1 + 3 × 101)~5- I 98 X 101 -1 = 012830 &c.
It is obfervable that when r is 1, the value of
r+I
I
m+2zn
I
is × L.?
•
2 rti.n
I
2n
m+2zn
m
; which may facilitate the compu-
m
tation.
52.
form
Let the equation propofed be x³ — 7 x + 6 = 0, to find the fum of the fractions of this
3x²+5
4x+1
The fractions will be
denom. produce
when for x is written each of the roots fucceffively.
+
3a²+5 3B²+5 3x²+5
4α+ I 46+ I
47+I
+
which reduced to a com.
48 .
a B y X a + B + y + 1 2 ×
2
2
2
2
a² ß + a² y + B² a + B² y + y² a + y² ß + 3 ×
+ B² + z² + 80 × aß + ay + By + 40 × α + B + y + 15, the numerator;
and 64. a B y + 16 × aß + ay + By + 40 × a + B + y + 1, the denominator.
Now compare the equations, and it will appear that p = a
+ B + y = 0, 9 =
aß + ay + By = −7, r =
a² ß + a² y + B² a + B² y +
fum will be
a By = −6; alſo a² + B² + y²
=
2
γα
2
y² a + y² ß = p q − 3r = 18.
p² − 2 q = 14, and
Confeq. the required
12. 18 +3.14 + 80.7 +15
64.-6+16.-7+1
287
495
53.
2.4.6.8(z)
Required the fum of z terms of the feries
3.5.7.9(x)'
Comparing this with the formula, we find p-1, r = 1; hence
x x
2x+1
(S) = √
=S=
x
=
S √
I
NI
2
or
S
2x+I
x
the whole fuent of
x x
I
2
or greateſt value of x, being = 1. If z be infinite, (S) = √c, rad. 1.
If
OF
211
SERIES.
If the ſeries propoſed be 1.3.5.7 (x)
2.4.6.8 (z)'
22
(5) = √ √
=s
x
√ √
S
x
>
I
or
we have p = -1, r = 1; whence
4
C
Ї
22
x
I
; the whole fluent of
being the circumf. (c) of the circle, of which the rad. is 1.
Hence it appears that the whole fluent
22
of
2.4.6(x)
3 · 5 · 7 (z) ³
22
x x
of
is =
I
*
1 . 3 . 5 (
(x)
2.4.6 (z
X
C
and
4
; and when z or the number of factors is infinite,
S
I X
22
λ' x
will be equal
the ult. value of
equals
f. 2.
2.4.6 (z
3.5·7(z)
I-
2x+
and the ultimate value of
I • 3 • 5 (x)
с
>
2 4. 6 (x)
4
But
2.4.6(z)
to any finite number of terms is
3 · 5.7 (z)
1.3.5 (z)
2.4.6(x) I
2.4.6 (z)
evidently =
X
I
22+I
and therefore when z is infinite
2.4.6(z)
3.5.7(z)
is =
X ; hence
2 Z
2.4.6 (z
1.3.5 (≈)
I
X
=
2 Z
1.3.5
5 (≈)
2 4.6 (z)
C
4
"
X and
2
X
I • 3 • 5 (≈)
c = the ult. val. of
2².42.6² (z)
1² . 3² • 5² (z) Z
6.28318530.
By affuming different values for p and r, may the ultimate value of a great variety
of expreffions be found by this formula; which is alſo of great ufe in the finding of
fluents, and in computations relating to curves formed by the fufpending of perfectly
flexible bodies, as the Lemnifcata, Lintearia, &c. and likewife to the elaftic curve,
the deſcent of a heavy body in the quadrant of a circle, and other phyfical enquiries.
To exemplify its ufe in the finding of a fluent, let that of
Comparing this expreffion with the formula, we find p₁ = −1, r = 1, z=n; hence
S
S
P²
x
I
HIN
• *
४
2
Hjel
1 . 5 .. 9. 13 (≈
3.7.11.15 (≈) *
P
+
16
8
4
27
x
x
be propofed.
I
But
S
X
is known to be
I -
where P is the periphery of an ellipfis, of which the axes are
8 and 2, and C the circumf. of a circle, rad. 1, confeq. the whole fluent of
27 - 12/04
x
1 · 5 · 9 (n)
is =
3. 7. 11. (x).
× & P + √ √ P² — C..
16
}
The
212
THE SUMMATION
The feries of the reciprocals of 5, 6, &c. factors are omitted; alſo the ſeries of
fractions of which the numerators are in arith. prog, and denominator 4, 5, &c.
factors; the ſeries of fractions of which the numerators confift of two factors in arith.
prog, and denom. 4, 5, &c. factors; and thofe feries of fractions of which the terms
are multiplied into a ſeries of powers of a given fraction. Theſe being all thought
unneceſſary, as they feldom occur in the folutions of problems; and more fo, as they
may each of them be eafily obtained by the methods laid down in the former part of
this treatiſe.
Thoſe that would ſee more of the application of the doctrine of feries to the folution
of problems, may confult Vol. I. of Ifaaci Newtoni Opera omnia, by Dr. Horfley;
in the Appendix to which (Logiſtica Infinitorum) are alſo given ſome uſeful formulæ
for expediting the arithmetical computations in feries.
Promifcuous Series which have been fummed by the preceding Formula.
2.4.6 4.6.8 6.8.10
200. (F)
+
+
+ &c.
3
3²
33
z + 2
(S) 81 × : 1
x +3
Z+I
≈ +3
x+1
X
ZX
2
2
3
3+2
+xx
X
2
z + 2
x+4
3.
*
3x5
3.5x7
201. (F)
+
+
+ &c.
3.5.7
(E)
א
1
ZX
+ I
48
2.3.5.7.9
3.5.7.9+ 2.4.5.7.9.11 2.4.6.7.9.11.13
5-18x²+24+7
768x°
15−44x²+44 x4
z = circ, arc. rad. 1, fin. x.
I
3
+
2.3. 5
2.4.5.9
3.5
2.4.6.7.13
S
which
>
202. (F)
(2) ** × S
X
X
Lucub. p. 146.
2304x5
**
3.5.7
2.4.6.8.9.17
+ &c.
may be reduced by Landen's
$
उक
× 32+ &c.
I
I
I
203. (F)
X+
ㄨ​ㄨ​+
2
•
3.4
L. 2 - //.
38
+
(ε)
204. (F)
(E)
205. (F)
(E)
1.2.3
23
I.2.
2.3
23
I. 2. 3
H
x+
2.3.4
38
94
2.3.4
풀
​3.4.5
57
3.4.5
I
27+
57
3.4.5
I
× 18+ &c.
× 18+
Iठ
I
80
4.5.6
× 31 + &c.
206,
OF
213
SERIE S.
6
5
206. (F) —
2.3 × +2.
ㄨ​ㄛ​+
2. 3.4
7
X+
× F + &c.
3.4.5
207. (F)
(E)
19
x+
•
1 2.3
28
2.3.4
39
X 27+
6
X + &c.
3 · 4 · 5
(E)
2.
208. (F) b+
8/0
+2
x3
b
262 363
+
+ &c.
464
(ε)
L.b-x
(S)
22-I
3
S 3,
209. (F) 1 + A + B + C + / D + &c.
(E)
I ‡ ¿ §
(S) 2−2%−2 × s,
x = 1002, (S) =
210. (F) + A + B
} ‡ §
2.
To
the fum of z—
the ſum of x-2 terms, s being the x+1th term. If
1·643818.
8
오
​+ C + D + &c.
the fum of 2-1 terms, s being the zth term of the pro-
pofed feries. If z = 101, (S) = 1194′2782.
I
211. (F) 1+
32
A+
2.3
4.5
52
B+
6.7
C+ &c.
(E)
the circumf. of the circ. rad. 1.
(S) 1°5707963—sX:☹−3+ +
4
140
15* 35xx 21x..x 33x.x
&c. the
3
ſum of z— 1 terms of the propoſed ſeries, s being the ath term, and x = 22.
If z = 1001, we have (S)
1'5529544.
12
20
+
+
I
2
I
212. (F) +
3
3.5
+
(E)
circumf. of the arc. rad. .
2.3.5 2.4.5.7 2.4.6.7.9
+
+ Sic.
I
213. (F)
+
3
3.5
+
+
+ &c.
(E)
I
214. (F)
I
+
I
+ &c..
(E)
215. (F)
2.3.7 2.4.5.9 2.4.6.7. II
circumf. of the arc. rad. .
m.m+n.m+2n (r)
I
m+n.m+2n.m+3n (r)
m.m+n.m+2n (r− 1). n. r− i
I
I
+
+
+ &c..
I. n+2
3.n+4 5•n+6
(E)
214
THE SUMMATION
12 + 8
(E)
6.n+2
11 - I
+
+
•
1.3.5
N— I · 1-3
6.1.3.5.7
n~ 1. n−3·125
+ &c.
8.1.3.5.7.9
1
I
I
216. (F)
n + 14
+
+
1.n+2
n-2
(E)
•
12 12+2
+
6.1.4.7
Where n may be any odd number.
+
4.n+5 7.7+8 10. n+11
Where n may be any number fuch that n+1 may be fome multiple of 3, or
of any number in the progreffion 2, 5, 8, 11, &c.
I
+ &c.
n-2.n−5 n—2n−5. n — 8
9.1.4.7.10 12.1.4.7.10.13
+ &c.
n.n-I
217. (F) 1-3 n + }
2
n.n-I · n - 2
2.3
(n+1).
(S)
2.4.6 (2n)
218. (F) u+
(E)
219. (F)
(S)
Τ
x + a
v「 ༤。
الله
+
+
1.3.5.
น
3
2
+
й
4°
SXL. 1-1.
a
น
x+a.x+b
I
+
น.
2n-I. 2n + I
2n+I
4
น
2
+ &c.
a
bx
220. (F)
x+a
x+a
(S)
221. (F) I + +
a
n
(E)
ab
a b c
+
x+a•x+b.x+c.x+d(x)•
x+a.x+b.x+c
abc (z)
x.x+a.x+b. (z+1)
x+b
+
cx²
d x3
x+a.x+box+c+x+a.x+b.x+c.x+a(x).
x+a.x+b.x+c (x)
I
a. atr
a. atr
atr.a+2r
+
+ &c.
n.ntr
n
n+rint2r
n-r
n—a+r
atr.b
222. (F) a+b.a+b+r
+
a+2r.b²
+
a + 3r. b³
·+&c.
a+b.a+b+r a+b.a+b+r.a+b+2r
a+b.a+b+r.a+b+2r' a+b.a+b+r.a+b+2r.a+b+3r
a
(E)
I
a+b
223. (F) 1 -
- 3 + 3
(E)
I
I
I
2
+
2
4+
S
+ &c.
66
224. (F)²
(E)
S
8
*
X x.
Ι
TI
} + 5 + 1 x ³ − x² + &c.
S
TI
x x
225. (F)
SERIE 5.
215
&c.
225. (F) {*³ – ‡*' + ǹ*" - ♬×" + 75 *¹9 —
(ε)
S***
226. (F) x − ½ x³ + † x³ -
{ ** + } ׳ — { x² + + +³ — &c.
(E)
S
x
I
When the figns of this feries are pofitive, the fummation cannot be exhibited by any
finite expreſſion whatſoever. And when is very nearly equal to I, it converges fo
flowly, that the common formulæ for this purpoſe are of little fervice. Mr. Maferes
has therefore lately inveſtigated an expreffion which gives a very near value of the
feries in this cafe; the arithmetical computation whereof he effects by means of a
property of the logarithmic curve. See Philof. Tranſ. Vol. LXVIII. art. 41.
227. (F) x +&{ x² +
5.7
4.6
x³ +
5.7.9
4.6.8
x++
5.7.9.11
4.6.8.10
*³ + &c.
I
x
(E)
X
I.
3
I
228. (F) 4x+
1.5
5.7
x² +
4.6
4.6.8+3+
5.7.9
4.6.8.10
** + &c.
(E)
I
3x
S
x
xx
+
x.
emler
I
4-
or, (E)
229. (F)
(E)
x
5
+
230. (F) ‡*+
(E)
X
5
4 X 7
I
2x X I-x
x² +
5.7
4.6x9
紅光
​3+*
I - X
+
5.-7
I
3*
x*
x
र्
४
5
*² +
23 +
4X9
4.6 X II
S
I
3 xžu
2
231. (F) +5 +256 77
(E)
4.5
4.6.7
I
6 **
लोला
makes
• *
5.7.9
-2.
4.6.8 x II
** + &c.
5.7.9
4.6.8 × 13
** + &c.
+
5.7x3
4.6.8.9
+
I
ΙΣ
5.7.9+*
+
4.6.8.10.II
S
S
xx
+
I
Hh
I
+ &c.
232. (F)
216
THE SUMMATION
1
232. (F) 2.4 X 5-7
(E)
1
18x
S
+x
5.7
x³ +
5.7.9
2.4.6.8.10X11.13
x²+&c.
5
2.4.6X7.9
xx
2.4.6.8 × 9.11
12x
J
Ι
+
3
I
Σ
I *
x2
xž x.
3
I ---- X
紅​醬​+
36 xže
I
+
I
05, (2) → S * ƒï+ * ƒ ==+¯***
or,
24 x
x
I X
1
233. (F) 1 + x + } x² + − x³ + 3 x² + &c.
+} 7
(E)
x
इ
234+ (F) 2-3 + 3 = 3 + 3-7
(E)
235. (F) x
(E)
or, (x)
1.3
1
X
*.
ल
x
I
XL.
2√x
I
1- √x
4 23
5 x¹
+
+ &c.
5.7
7.9
3x-1
I + √ X
81x
XL.
+ 1.
I
+
I
2
3 x3
2² X 2 X 7 x7
2³ × 2.3 × 11 *
LE
+ &c.
I+**.
SAN
*
Hyperb. arc. x being the abcifs to the aſymptote.
3
2.4.5.2
Circ. arc. rad. 1. fin. ž.
1
C
* + x +
I
236. (F) +
+
+
S
2.3.2
3.5
2.4.6.7.27
3.5.7
+
2.4.6.8.9.20+ &c.
(E)
I
I
I
I
237. (F)
+
+
+
+ &c.
I.3 I.2X4
1.2.3×5
I.2. 3·4X6
(E)
。
I
2
2
238. (F)
+
1. 7. 11. 15
5.9.7.19
3.11.17.23
(E) 4.
4√ 2x
HT
I
४
2x² +
X
I + x
I
+ &c.
7. 13.5.27
I
I ΑΤΣ
37
I+*
9240
or, (2)
72 circ. arc. r. ✔¼, t. §−64 arc. r . √ž, t. 2 – 16 arc. r. 1, t. I,
239. (F) Circ. arc. fin. 8-3+ arc.s.
(x)
32
√2
XL.
2 ✓2
2+ √2
2.3
Circ, arc, rad. 1. fin. .
32 L.2, +
√15-48
3.4
2041
3080
+ arc. s.
√24-√15
4.5
+ &c.
240. (F)
OF
217
& ERIE S.
I
.I
+
+
+ &c.
m+ 2n
m+3n
m+4n
I
240. (F)
(ε)
m
m+ n
ر
ˇ
I
a.a+1.a‡2 (m)
241. (F)
I
(S)
X:
m-I
I
+
a+1.a+2.a+3 (m)
+ &c.
I
I
a.a+1.a+2. (m− 1 ) a+z.a+z+1 (m−1)
I
a.a+1.a+2 (m−1)
(ε)
242. (F)
n+2 2.2+3
I
m
I
I
I
+
+
#
+ &c.
3.7+4
I
I
I
(E)
X :
+ = +
j
n+I
#+I
n
n I
— &c. (n+1).
I
I
I
243. (F)
+
+
+
+ &c.
n+ 2
3• n+4
I
(E) X : +
#+I
244. (F) 2. x+3
(E)
245. (F)
2.n+3
I
n+I
7--2
x
n+2
› n.
+
I
I
5.n+6
1-2
I
7.n+8
+ + &c. ("+¹).
2 A-4
4.n+5
I
+on+y
2
I
+ &c.
I
+
1 - 1 + 1 = 3 + 1 = 5 + &c. (" +
').
2
**+4
+
&c.
X :
n+ I
1-5
1+ (3)
n+ I
246. (F)
(E)
× : **
x+2
+
n+2
#+ I
is
+3
x8+1.3
I
*"+6
3.n+4 5.7+6
x
X
・S-(+)
+
+ &c. {
I+
n
n 2 22-4
n
+L. Vit
**+4
+
3-x+4 5.n+6
*
+
(1)
x
+
+ &c.
+
n+1.n x+1.8‡a ' h+1;n=4
±{L.1−x².
+ &c. +
I
+
#+
لة
Hh 2
247. (F)
215
THE SUMMATION
247. (F) x +
M x3
+
3
(E)
248. (F)
(ε)
249. (F)
x
m
x y r
m
X
m.m+1.x³ m. m+1.m+2. *7
+ x
+ *
४
EIL
2 X 5
S=
211
x-y
I+
I +
x
y
X
•2
I-
+
M3
y + x
m
jr + x
178
y r
+
113
y
2.3X7
+ &c.
y² +
1³ + &c.
4m 3m
m-
yr + x
r
yr + &c.
3m 2m
聶
​m and r being affirm, integers.
y
3
27 +27 +7
+
y
*
2. m
+ &0.
3m
y
+
+ &c.
x
(E)
x
x-y
y
m and r, as before.
{
Theſe two theorems are particularly ufeful in computations by the Method of
Refiduals.
?
250. (F)
+ +
I
2
{ *
дов
+
22
2
(S)
X
X
1-r
I
+
710
+ &c.
52
25ì. (F) a + b + 6 + d + e +ƒ + &c.
() = + + + +
á-b
+ &c.
a-b
:
Here a = r²ac, a = r² a − j² à + e, ä = µő
дов
Am gut α of gr²
a
g, &c. allo
b
b = r² b−d, b = r² b−7' b + s, &c. and r=,=, &c
252. (F) a − bx + c x² - d x³ +ex4
a
&c.
(E)
a
I+*
dx
dx²
+
2
+
+ &c.
I+x
I+x
IN
Where'd a-b, a = a -2b+c, ä = a−3b+3c−d, &c.
d
253. (F)
OF
219
SERIES.
253. (F) a + bx + c x² + d x³ + ex² + &c.
(2)
a
254, (F) 2 X:
(E) *. L.
52
255. (F)
m
I
1.2
dx
ä x²
&c.
I X
+
+
3.4 5.6 7 8
+
1+*
I-X
I
.L
$4
+ &c.
+3m +
1-XX
+
ܕܨ
2. 3 m³ 3.5m³ 4.7 m²
2 m−1
1 × L.2m—1-2mL.m
+ &c.
(E)
256. (F)
(E)
m
x
257. (F)
+ +
4
7
(E)
or, (E) / L.
I
$3
+
+
3m³
5 m³
+
7 m²
7
+ &c.
IO x13
M
+ &c.
I
{ L.2m-1.
+ +
10 13
Si
— + ¿L.1 +*+** + § circ. arc. rad. ✔‡, 1 .
t
。2n+n²
++
258. (F)
I
e4r+#2 I
6n+n2
I
+
+
+ &c.
2
+
¿ ³ + "
I
*
1 + 1/2 x
+
(E)
+
+ + + &c.
&c.——
259. (F) x +Ţ²² +ò²‍+7
x+3
(S) 1 -
I
Z
I
- | =
ย
+ &c.
I
I
260. (F) 2/3 V2 + 24-3
2√2.3
+
2
2+1
+
4√ 3.4
2√5-√4
8√ 4.5
+ &c
I.
(E)
√2
I
261. (F)
I
+
+
2.3 × √ + √
3·4× √ 3+ √
4•5 × √&+✔}
(ε)
+ &cr
As
220
SUMMATION
THE
As thefe laft fpecies of feries feldom admit of any finite expreffions for their fums,
the general method is therefore by approximation; which may be always effected by
the theorem in Order 48. or by N° 251, 252, 253. when the terms of the propoſed
feries are reduced to their fimpleft form. And to thefe feries is alfo Dr. Waring's
Nova Methodus Differentiarum at the latter end of his Medit. Analytica particularly
applicable.
It may not be amifs now by way of conclufion, just to point out to the reader the
various proceffes whereby the fummations of feries are generally obtained, that in
cafes where he may not fall upon a method of reduction to any of the preceding
fo:ms, he may avail himself of them in applying one or other of the modes of invefti-
gation. The Bernoullian method, which is the moff fimple, is performed by af-
fuming a feries fuch, that being fubtracted from itſelf, the remaining feries fhall
coincide with that propofed to be fummed, by tranfpofing the negative terms.
De Moivre's method confifts in multiplying an affumed feries by fome binomial or
trinomial expreffion, which is then equated to o, and the negative terms tranfpofed.
Theſe two methods are illuftrated in p. 91, &c. and for other examples fee
De Moivre's Mifcell. Analyt. lib. 6. cap. 3. or Waring's De Infinitis Seriebus,
p. 389.
Another method of fummation, which is a very extenſive one, is by aſſuming a
feries which is compofed of the powers of an indefinite quantity and known coeffi-
cients, and of which the fum is exhibited by a finite expreffion. Both fides of the
equation are then multiplied by fuch an affumed expreffion, containing the fluxion of
the indefinite quantity, that may be conducive to reducing the affumed feries to the
fame form with the propofed one, by taking the fluents of the terms; and this opera-
tion may be repeated by other affumed expreffions, if the coefficients of the propofed
ſeries confift of two or more factors. Various examples of this method may be ſeen
in the before-mentioned authors. The folution of the 13th question in the Ladies'
Diary 1780, by Mr. Willian Sewel, is an excellent illuftration of this procedure.
If the propofed feries contains the fucceffive powers of fome given quantity, the
fummation may often be elegantly deduced by multiplying refpectively the terms of
the feries by thoſe of an affumed one which confift of the correfponding powers of an
indefinite quantity, and equating the whole to another indefinite or unknown quan-
tity. Then taking the fluxion on both fides the equation, and tranſpoſing, there
ariſes a fluxionary feries, the fluent of which may generally be exhibited in a finite
expreffion; and therefore the value of the indefinite quantity to which the feries was
equated become known, and of courſe the value of the propofed feries in finite terms
by ſubſtituting for the indefinite quantity fuch a value as may reduce the feries of
which the fluent is taken to that propoſed to be fummed. By this procefs are the
œcumenical formulæ inveftigated in the former part of this treatife. The Rev. Mr.
Crakelt
}
OF
221
SERIE
S.
Crakelt has alſo made ufe of a fimilar method in the folution of queft. 55. Brit.
Oracle, p. 240.
There are various other methods of obtaining the fummations of feries; as Stirling's
Differentials; Taylor's, Simpſon's, and Emerfon's Increments; the latter of which
is very extenſive, and applicable to almoft all orders of feries, but very operofe in its
computations. And in the works of the authors I have mentioned in this treatiſe,
may be met with various fingular and peculiar proceffes for the fumming of certain
kinds of feries, all which are well worth the reader's perufal, if he be defirous of being
thoroughly acquainted with this extenfive branch of the analytic art.
FINI S.
ERR AT A.
Page vi. 1. 22. r. becomes, p. ix. 1. 2. from b. r. neque, p. 15. 1. laft, for obferving
to correct the flucnts if neceſſary, r. the fluents being rectified by ſubſtitution, p. 23. 1. 18.
r. logarithms, p. 26. for L. 1+x, r. L. 1+x, alfo for L.-2, r. L. 2—, p. 42.
3
1. 2. from b. dele pairs of, p. 43. 1. 2. for A, r. F, p. 44. l. 10.
19
in the expreffion
M, for
Sπ, X. NSF, p. 50. 1. 10. dele SS p. *64. l. 11. r.
,
108
I-y.y'
12
1+ y¹2
9
and 1. 6. from b. and the following lines, for n, r. n, p. 79. 1. 6. from b. for
a
rad. ——, 1.
2
>
r. tang. —, p. 89. 1. 4. for 1 in the numerators, r. progreffively, 1, 2, 3, 4,
2
&c. p. 93. 1. 4. r. terms, p. 96. 1. 11. for fum, r. fummation, p. 148. 1. 6. from b.
for even, r. whole, alfo in p. 149. 1. 9. r. whole, Example to N° 39. in the denom. of
the 4th term, for 1 r. . Sheet M is by miſtake paged as the preceding one, the
pages are therefore diftinguiſhed by an aſteriſm in the reference.
I I
I 3
As Mr. Lorgna has not reviſed his Treatife, it may be neceſſary to point out the
principal typographical and other Errors therein, that whoever has the Curiofity to
compare the Original and Tranflation, may be apprized of th eCorrections.
Page 23. Examp. 4. the figns of the propoſed ſeries ſhould be alternately + and
p. 32. 1. 7. for x, r. 1, and 1. 9. for p+49, г. p + 29, p. 34. l. 15. г. n s π, p. 41. l. 3.
x
√
r. 1 in the denominator, p. 50. 1. laft for 1-z³, r. 1-y³, p. 52. 1. 8. for 2 §
r. 2, p. 55. the expreffion C fhould be as in art. 55. of the tranflat. p. 67. the
numerators of the terms fhould be progreffively a+b, a+2b, a+3b, &c. a+zb,
3
P. 72. 1. 12. for
, p. 77. 1. 10. for p+39, r. p+29,
2x+3
2
I
r.
and
2x+2 2x+2
T
p. 82. 1. 11. for 8-10%, r. 8 + 10%. In Examp. 4. p. 83. π is equal, which is
omitted, p. 87. in the expreffion Q for qns♪ r. qnsæ, p. 91. 1. 2. r. É, É, alſo in
p. 92. r. M, M, p. 96. 1. 14. for 9+29, 1. p+29, p. 97. in Q for t+u, r. t+uz.
The fums of the following feries will be found as in the tranflat, Example 2, P. 47.
Ex. 3. p. 48. Ex. 4. p. ib. Ex. 5. P. 93.
SUPPLEMENT
то
PROFESSOR LORGNA'S
SUMMATION OF SERIES.
TO WHICH ARE ADDED,
RE M
ARK S
ON
MR. LANDEN'S OBSERVATIONS ON THE SAME SUBJECT.
BY THE TRANSLATOR OF THE ABOVE WORK,
HENRY
CLARK E.
DAMNANT QUOD NON INTELLIGUNT. QUINTILIAN.
LONDON:
PRINTED FOR THE AUTHOR; AND SOLD BY J. MURRAY,
No 32, IN FLEET-STREET.
MDCCLXXXII.
:
7
!
1
+
?
;
1
→ }
[iii]
U
ADVERTISEMENT.
PON revifing my tranflation of Mr. Lorgna's Treatife on Series im
mediately after its publication, I found that ſome additions and alte-
rations were neceſſary to render it complete. Alſo, that in the Appendix,
I had omitted ſeveral curious and ufeful formulæ, fome of which were ori-
ginally intended to have been inferted therein, and others that have ſince
been inveſtigated. Not that any other principles have been diſcovered,
befides thoſe pointed out in the Appendix, whereby this branch of the
Mathematics may be farther extended; but only the application of certain
and peculiar proceffes therein. Indeed no new principles feem to be want-
ing (fuppofing any could be found) for, not to mention the well-known
methods peculiarly adapted to this purpoſe, every one knows that a num-
berleſs variety of feries with their fummations will be obtained by barely
expanding fractional or radical, binomial, trinomial, &c. expreffions; which
being again differently combined, produce a variety of other feries. It is
alfo as well known that by the application of the fluxionary calculus to this
ſubject we may produce an infinite variety of feries that are fummible; the
only addrefs or artifice requifite herein, being to affume fuch expreffions
that when they have undergone the various operations of multiplying, &c.
by the fluxion, or other functions, of the indefinite quantity, neceffary to
reduce their expanded values to the form of the feries propofed, the fluents
of the refulting expreffions may be attainable without infinite feries; that
is, either algebraically, or by means of the conic fections.
I had intended, in this Supplement, to have given the inveſtigations or
demonftrations of the formulæ for fummation in the different orders of
feries in the Appendix, under the names of their refpective authors; but
finding it would far exceed the limits I had propofed, and the utility thereof
not appearing very great, as the names of thoſe authors were before fpeci-
fied, I have, at prefent, relinquished the defign.
Τα
iv
ADVERTISEMENT.
To offer any apology for the typographical errors which have eſcaped in
this Treatife, would, I apprehend, be needlefs; as every reader must be
acquainted with the difficulties attending the corrections of the fheets in a
ſubject ſo intricate to a compofitor, when the author's fituation or diſtance
from the prefs renders it impracticable for him to inſpect the reviſes. Some
apology however may, perhaps, be thought neceffary for the freedom I
have uſed in this Supplement refpecting Mr. Landen's Obfervations, &c. In
defence of which I can only fay, that it did not ariſe from any private
pique or perſonal refentment towards that Gentleman, for the treatment he
has thought proper to fhew me on this occafion; but from a full and clear
conviction of the truth of what I have endeavoured to vindicate. And as
truth has been my object in this endeavour, I would wiſh every reader who
may not fully comprehend the fubject, not to be implicitly led on the one
hand by my bare affertions, nor on the other by the ſuggeſtions of thoſe
whom envy or prejudice may have prepoffeffed; but to fufpend his judge-
ment till he is thoroughly acquainted with the principles, and can deter-
mine for himſelf. For, as all men are liable to error, I would not, by a
too prefumptuous confidence affert, that all I have advanced on this fub-
ject is indiſputable; I fhall therefore be always ready, on conviction, to
acknowledge my miſtakes, and fhall welcome truth whatever quarter it
may come from. After all, let not the words of Horace be forgot,
Sunt delicta tamen, quibus ignoviffe velimus.
A SUP-
( : )
A SUPPLEMENT, &c.
+1
9
x
262. (F)
p + q
(E)
1. When
9
+1
१
+2
p+29
x
P
?
+
x
9
+3
p+37
•
It
*
+4
+ &c.
+49
is a poſitive whole number, and the figns of the feries are +,
I
fum is expreffed by
X :
L.I
I - X
q
1
P
x
x
9
x
&c.
2
3
4
P: 9
the
2. If the figns be + and - alternately, and a pofitive whole number, the ex-
૧
I
preffion for the fum becomes
×:±L.1+x=*
q
P
9
Н
2
+
the upper or under fign obtaining according as is even or odd.
+3
9
+1.
+ &c. +
3
p: 9
3. When x is 1, the fum in the foriner cafe will be infinite; but in the latter
I
it will be X: ±69314721 ± ± &c.
q
I
P: 9
4. And here we may obferve, that the formula to N° 271. inclufive, are each fup-
poſed to be generated while x from o attains the value denoted by X; from whence
the corrections that may be neceffary in the formula (E), when they are not purely
algebraical, will be obvious.
B
263. (F)
2
A SUPPLEMENT, &c.
11
+2
x
n
+3
263. (F)
(2)
1
+1
I
12
X
p + q. m + n
+1
I
X:
ng
+
9
x
π
When π
T
(
m
P
n
9
J
p+24.m+2n
x
P + 39 · m + 3n
I
± &c.
- x x S t + St
is an affirmative whole number, and (X), the ultimate value of
π muſt be an
* fuch that X* =1, the expreffion is reducible to algebraic quantities, the figns of
the feries being +. If the figns of the feries be alternately and
+
See theorem p. 18, when x is 1.
even pofitive number.
+ I
S
x
264. (F)
(.Σ)
When π
p+q.m+n.rts
I
ngs.π-w
X.:
+1
+2
X
p+2q.m.+2n.r +25
9
I
x x
+3
x
+
+ &c..
p+ 39. m + 3n.r +35
111
x f S x S S
T
+ *
It
1) and w G
TIX
m
n
:)
W
x
I
X
FX
are affirmative whole numbers, and the
9
XT χω
I
I
the expreffion is algebraical; the
T
W
Π
W
value of X ſuch that
figns of the feries being affirmative. If the figns. of the feries be alternately
,
™ and . muſt be even affirm. numbers. See theorems p. 42. when x is = 1.
and
265, (F)
x
-t
ti
น
p + q. m + n. r+s.i+u
H
+2
น
x
p+29.m+2n.r+2s.t+24
+3
น
S
x
X:
(2) 1x
IF X
-1:
ngs u
W.T
f
+
± &c.
P
P + 3 q. m + 3 n.r + 3 s.1+3%
X*
9
X
x
1
T. T
...W
*T
f
t
น
χω
+
7.7
一起
​W
ย
IF X
X♪
d.d-wiπ-
..77
n
14
I
IF X
W
m
1. When (-)-(-5) (-5)
T
น
น.
น
nk
•
IF *
are pofitive whole num
bers,
A SUPPLEMENT,
&c.
3
bers, and X is taken of ſuch value as correfponds with the equation
X*
χω
+
π π
•
I
W
}
d.d-w.π
- บ
>
the expreffion will be algebraical; retaining the upper figns. If
♪
the under figns be taken,,, must be even pofitive numbers. See theo:ems p. 62,
63. when x is = 1.
In particular cafes,
2. If be a whole pofitive number,
ע
a whole number, the figns of the feries
being affirmative; or even numbers, the figns being alternately + and; and the
≈
T
relation of the factors in the propofed feries be fuch that π- w = ♪.♪-w, the ex-
preffion will be algebraical though and ♪ be fractions in the former cafe, or uneven
numbers in the latter; provided x be ultimately taken of fuch values as arife from the
above equation. In thefe circumstances the formula (R) p. 63. becomes
+
r+25 r+35
I
X:
nqu. ய
W
- W
I
r+s
I
I
+
= (w)
(E)
I
I
I
I
+
X :
+
+
± (~~d).
nsis. ZZ
•
T
p + q 4+29
p+39
3. If π
be a whole pofitive number, w♪ a whole number, the figns being affirma-
tive; or even numbers, the figns being + and -; and w. T
ய
? — *
π-w=d.8-d, the
fum will be expreffed algebraically though and ♪ be fractions in the former cafe, or
uneven numbers in the latter, if the ultimate values of x be fuch as correſpond with
the above equation. Under thefe limitations the above-mentioned formula becomes.
I
I
I
I
x:
+
น
nsu.π.
P+9
(x)
+
± (*)
p+29 户​+39
I
I
I
I
+
n q
น
X :
•
T w
•
+
r+s r +25
+
± (a: —d).
+35
172
+I
+2
a+b.x
-+3
n
a+2b.x
a+3b.x
266. (F)
+
+
+ &c.
p+q. m +n
p+2q.m+2n
p+39. m + 3n
P
b
9
X
(ε)
X: xX*
+:
X
T.
q n T
S
x
1. When ≈
(
9
) is a poſitive whole number, the figns of the feries being
affirmative; or an even pofitive number, the figns being + and
B 2
>
the expreffion
will
4
&c.
A SUPPLEMENT,
will be algebraical, provided X be any value in the equation X =
a
Р
is =
See theorem p. 68.
b
9.
X
where x
2. If x― be a negative quantity, and an uneven pofitive number, and the value
7*
of X fuch as may correfpond with the equation X” =
16
X
, the fum will be alge-
x
braical when the figns of the feries are alternately + and -. See theorems p. 72..
when x is = I..
+1
+2
+3
a+b.x
a+2b.x
a+3b.x
267. (F)
It
+
± &c....
p+q.m+n.rts
p+2q.m+2n.r + 25
p + 39 m + 3n.r + 35.
m
b
Κ
Xd
9
x
x
x
(E)
T-X. X
n
น
x
x.
X
+
q n s T
d
=
IF X
x
I + x
1-8
X
x
1. The fum will be algebraical when
(
m
n
IF X
9
), and ♪ ( − 2)
r
S
are affir-
mative whole numbers, and the value of X is fuch as correfponds with the equation
See theorems: p. 82. when.x is I..
H
Xd-
x
+
+
бот
d
♪➡ T
Particular cafe.
T-
2. If be a pofitive whole number, x, and -x or ♪ be negative, though -
♪ and (confequently) - be fractions, the fum will be had algebraically when x is
any value in the fame equation, that is in X" = 1, the figns of the feries being affir- -
mative. Alſo, if be an even number, the fum will be had in algebraic terms when
the figns are + and, though and (confequently) - be odd numbers, the reſt
being as before. But it may be obferved, that the feries do in ſuch caſe become eſſen-
tially of the form in Nº 263. the terms being multiplied by the conftant quantity.
a+b
rtis
= +2
268. (F.)
a+b.ctc.x
p+q.m+n.rts
+
a+2b.c+2e.x
P+29.m+2n.r+25
+
+3
a+3b.c.+3ex
P+39+m+3n.r+35,
± &c..
(E)
A SUPPLEMENT,
5
&c.
(=).
be
n s
q n
X
x x x *
x
κλ
π
K
ż
dod-
πολη
б
π
X
IF X
λεκ
مین
1. This expreffion becomes algebraical when ♪ and d—are pofitive whe num-
bers, the figns of the ſeries being affirmative; or even numbers, the figns being +
and -; and the ultimate value of x fuch as arifes from the equation
κλα
%8
X
7
π
=
7
2. Hence, in particular caſes, if we affume ♪ = λ, then muſt be a poſitive whole
x X".
number, and ≈ a value ariſing from the equation X” =
If we take x, then muſt ♪ be an affirm. whole number, and xa value in the
equation X³ = 2—♪♪.
λ
If bea, muft be a pofitive whole number, and a value in the equation
x =
X
=
x
If x = ♪, « muſt be a pofitive whole number, and x correfpond with the equation
X**
=
λ
If, d, be even numbers, the algebraic fum will be had when the figns of the feries
In all theſe cafes, X muft evidently be lefs than 1.
are + and
›
3. But when the figns of the feries are alternately + and -; and x-, or -
negative, as alfo x- or a-d; then muft and ♪ be odd pofitive numbers, and
a value in the above equation.
Th
4. Hence, in the firft particular cafe above, if we take = 2, the expreffion
will be algebraical, whenever is any add pofitive number, and x a vaiue in the
equation X = 1 (See theorem p. 97. when x is 1.). The fame holds good if
we take × = d, and ≈ = 22. In the fecond cafe, if we make 2, the expreffion
is reducible into finite terms, whenever is any odd paktive number, and x a value in
the equation XI (See theorem p. 97. when x is 1.). Alfo if, and
2x, the fum will be expreffed in finite terms under the fame reftrictions.. In
theſe caſes one value of X will evidently be always 1. It is alfo obfervable here,
that.
6
&c.
A
SUPPLEMENT,
that the feries do under the above limitations become effentially of the form in N° 266,
being multiplied by fome conftant quantity.
269. (F)
-+:
ab.c+e.x
p+q.m+n.r+s.ttu
+
+2
น
+1
a+2b.c+2e. x
p+29. m+2n.r+25.1+244
24
+3
a + 3b.c+3e.x
p+39. m + 3n.r+zs.1+3u
± &c.
(ε)
be
q n s u
x x X²
q
X:
IF X
7"
T W
T
1
*
น
+
κλ
x
πω 17*
x-d.λ-d. Xus
d.d-π.w-d
+
•
•
४
I 7 x
7.
x
น
x
T
ба
IF X
1. This expreffion is algebraical whenever w, T, are pofitive whole numbers, and
X any value arifing from the equation
κ λ χω
+
πδω
*
+
π
δ
ω
7
d.d-π.w-d
T
= (x^2 +
κ
7
+
H
διδα που
d
=),
X--- W λ
•
W
ω
w.dow
>
the figns of the feries being affirmative. When
the figns of the feries are alternately + and -, w, π, d muſt be even numbers. See
theorems p. 107. when x is 1.
In particular cafes,
W
P
w
2. If the value of be negative (fee p. 106.), and equal to and be a
whole pofitive number, w- a whole number, the algebraic fum is attainable, though
w and ♪ be fractions, provided the ultimate value of x be taken ſuch as may correfpond
with the above equation, that is, with
κ λ χω
δω
T
T
=
; in which, if
π
we ſuppoſe X = 1, we ſhall have x=d, and w = λ, or x = w, and ♪ = ^. Under
theſe limitations the general formula p. 107. becomes
(E)
p + q
P զ
I
I
I
X :
+
+
± (*)
น.
Rs
I
I
I
+
X
+
น
u.
P
be negative, and =
W
8
3. If
R
p+29 +39
+ ± (w-d).
r+s r+25 r+35
; and be a whole pofitive number, w- a
whole number; the fum is expreffed algebraically though w and are fractions. And
fuppofing
A SUPPLEMENT,
7
&c.
P
I
X:
น.
p + q
(E)
I
fuppofing the ultimate value of x to be 1 in the above equation, that is, taking × =π
and aw, orxanda %, the fame expreffion becomes
+
+29 p+39
I
±
± (d)
Qn
I
I
I
+
น
X
W
+
m+ n
+
m+2n m+3n
(WIT).
4. The comparing of
е
R
and
w-3
requires ± ≈
= ♪
T
to be a whole number,
and when x is ſuppoſed to become 1, it is requifite that be♪; in this caſe there-
fore the fum is infinite when x attains that value.
a+b.c+e.f+g (k) x
W
270. (F) p+q.m+n.r+s.i+u (l+1)
+
It
a+3b.c+ze.ƒ+38 (k) x
a+2b.c+2e.f+2g (k) x
+2
p+2q. m + 2n.r+25.1+3u (1+1)
W
+3
p+39 · m + 3n. r+35 . 1+ zu (1+1)
± &c.
(=)
W
р
#
9
X
× AX
9
ก
+BX
+ CX
И
+DX
(1+1) x
I
X
X2
BXX:
+1
+
(1)
p+q
p+29
+39
I
X
X²
+
-CXx:
+
+
(pj
X
p+q
Þ+29
P+39
I
X
X2
-DXx:
+
+
(a)
>
1
A =
beg (k)
B =
qnsu (1+1)
p + q P+29 P+39
&c. to /terms.
X
Гаву
&c.
T σ
R
་
Y &c.
C=
&c.
"
a
В
BY &c.
про
a
α, B, Y, &c. =
P
P
P
£, &c. refpectively.
9
9
g
9
á, é, j,
Y
8
A SUPPLEMENT, &c.
(E)
ά, B, %, &c.
a
m
Sic. =
10
&c.
>
n
a
ä, &c.
11
&c.
b
S
m
P
π, P, σ, &c. —
11
9
9
P
&c.
"
9
P
m
*, &c.
11
&c.
9
n
%, &c.
P
r
-
&c.
q
S
mn
g
t
ข
n
10
The index is =
,or, &c. according as I is 1, 2, or 3, &c. and w, pɔ
c. are pofitive whole numbers.
a+b.c+e(k) x a+2b.c+2e(k) gdx² a+3b.c+3e (k) g·
8. 8-1
ď x³
2
271. (F)
p+q.m+n (l+1)* p + 29. m + 2n (l+ 1 )
+
a+4b.c+4e (k) g
2
8 - 2
3
d³ x4
+
p+49. m + 4n (1+1)
p+39 · m + 3n (l+1)
+ &c.
AX
p
q
+
XX
111
p+q.p+29(π) BX
n
•
p+9 · p + 29 (1) CX ~ =
89+p+29 •89+p+39 (*) d**89+p+29 •89+p+3? (p) de(²+1) ×
I
9
X:
p+q.p+2q(≈)B.1 −dX\8+±
89+p+29.89+p+39(x)X*~¹dπ p + q
p + q · p + 29 (1) С. 1 — dXs + I
I
8 9 + 1 + 2 9 • 59+p+39 (f) Xi-1 dp
&c. to /terms.
272. (F)
+
P
J
9
४
I-dx g
89+p+29.dX, 89+p+29.8p+p+39.d²X²
p+q.p+29
p+q⋅p+29 · p +39
9
X:
+
p + q
89+p+27.dX
p+q⋅ p+29
(P)
A, B, C, &c. `, p, σ, &c. as in the preceding theorem.
I
b-e+2.b+2.2b-g−2e+4.2h+8+4
I
h~e+1.b+1.2h−g−2e+2.2b+g+2
I
b-e +3.b +3.2b−g−2e+6.2b+g+-6
+1
(x)
(E)
+
Sec.
I
I
I
I
X:
+
+
8 + c • ½ g + se
2b-8-2e+2
2b-8-2e+4
2b-8-2e+6
(8+e)
I
I
I
I
X
80428 ¢
b-c+I
+
b-e+ 2
b-i + 3
(e).
1. When
A
9
SUPPLEMENT,
&c.
1. When the figns are affirmative, and g is any odd number (e being any whole
number whatever), « (+2) and 8 (2) will be fractions.
2. When the figns are alternate, g any number in the progreffion 2, 6, 10, 14,
18, &c. and e any number in the progreffion 2, 4, 6, 8, 10, &c. and will be
odd numbers. In the former cafe, w (g+e) will evidently be always a whole num-
ber; and in the latter, always an even number.
273. (F)
I
1
b-e+2.2b+8+4·2b+8−2€+4•b+8+ 2
+
I
+
b-e+3.2b+8+6.2b+g¬2€+6.h+8+3 be+4.2b+8+8.2h+8−2e+8.6+8+4
I
+ &c.
I
I
I
X
+
g+ e.g²+2ge
b-e+2 b-e+3
+
± (8+i)
b-e+4
(E)
I
I
I
X:
+
+ (e).
£8² e + g
2b+8-2e+4 2b+8-2e+6
w
1. What is obſerved in the laſt theorem is applicable in this; writing for, and
- for w.
2. If, in the preceding formulæ (No 265, 269, 272, 273,) the quantities -♪,
w-d, or w- 7, &c. ſhould come out negative, write one value for the other in the
theorem, that is for write the value of ♪, and for write the value of
the algebraic fum will then be truly exhibited in thofe cafes.
&c. and
b
C
274. (F)
A, s.
Nyi - b²
2
- a² + A, s
2
s.
√ r² - c²
-r
r
*
J
C
d
+ A, s
(1)
A, s. a, radius r.
+ &c.
a being any number not greater than r; and b, c, d, &c. any numbers decreaſing
from a to o.
ab+ √ y²-u²or²—b²
275. (F) A,σ.
2 2
bc+ √ r³-b²‚r²—¿²
2
cd + √ r²—¿².r²—d²
(E)
+ A‚o.
A, σ.a, rad. I,
+A,
+ &c.
276. (F)
A, t.
(E)
go2.a-b
+ A, t
12 +ab
A, 1. a, rad. r.
r² + b c
guz tab
277. (F)
A, T.
b-a
g² + b c
tbe
+ A,
r² + cd
T
+ &c.
6-b
d-c
C
+ A, T
a, b, c, &c. increafing from any value to r.
x² .c-d
+ A, t
+ &c.
g² + cd
a, b, c, &c. decreafing from any value to o
(E)
10
SUPPLEMENT,
MEN T, &c..
A
(=)
A, 7. a, rad. r.
a, b, c, &c. increaſing from any value to infinity.-
278. (F) A,/.
(E)
a b
√a²-1.6²-1+I
A, ƒ. a, rad. 1.
+A,S.
b c
√ b²-1.c²-1+ I
+A‚S.
c d
c²-1.d²-1+1
+&c.
a, b, c, &c. being decreaſing quantities from any value to 1..
a b
bc
c d
279. (F) A, s.
+A, s.
+A,s.
+ &c..
2
a²_1_√ b² - P.
(E)
·
A, s a, rad. 1.
280. (F)A,..
a, b, c, &c. increafing from any value not less than 1 to infinity.
a+b−2r.√ zar-a².b-r-√ 2br-b².a-r
r. √ zar — a² — r. √ 2br~ b²
-
+A, x..
r.√
b+c-2r.v2br-b2.c-r-V 2cr-c².b-r
r. √2br-b²-r. √2cr-c²
c+d-2r. √2cr-cd-r-V2dr-d.cr
+A,,.
+ &c.
r.. V 2cr ¿² —r. √ 2dr — ď²
(E)
A,›
a, rad. r..
281. (F)
A,v.
+ A‚v.
+A, v.
a, b, c, &c. increafing from r to 2 r..
r.a-b. v zra—a². √ 2rb-b² r.b—c. √2rb — b². √ 2rc — c
N
r-b.√ 2ra-a²+r—a.√ 2rb-b² r-c.√ 2rb-b²+r-b. √ 2rc-c
r. c-d.√ 2rc — c².√ 2rd — d'
r―d. ✓ 2rc- c²+r-c.√ 2rd-d²
=
+ &c.
(E)
A, v .a, rad. ra
a, b, c, &c. decreafing from r to o.
In the laft eight formulæ, A denotes circular arc; s, o, t, ™, f, 5, v, ", the fine,
co-fine, tangent, co-tangent, fecant, co-fecant, verfed fine, and verfed fine fupple-
ment, reſpectively.
+3
282. (F)
* +
(E)
3x5
2.3x² 2.4.574
Circ. arc, rad. r, fin. x.
+
+
3.527
2.4.6.7r
+ &c.
ро
c² x3
283. (F)
(E)
x +
+
2.3a+ 2 4.5a
+
•
284. (F)
* X: I+
(E)
9²
2.3
Parabolic arc; an ordinate,. q. =
12
Elliptical arc, a the femitranfv. c the femiconj. x the diſtance of an
ordinate (parallel to the conjugate) from the center.
•
8a+ c⋅ — 4a²c++ c°· 327
2:4 6.7a
+ &ca.
94
2.4.5
+
༢༠༠
2.4.6.7
3.598
2.4.6.8.9
+. &c.
x
(less than 1).
par.
285. (F)
A SUPPLEMENT,
&c.
285. (F)
(E)
CX: A+
LB-
92
393
C +
Ꭰ
2
2.4
2.4 6
3.59+
2.4.6.8
+ &c.
Hyperbolic arc (beginning at the vertex); a = the femitranív,
+
the femiconj. q= -, y an ordinate, A = L.³±x
a² + c²
axe,
c+
V x
2A
y³x -
B=
C=
3c² B
2
4
where x = √² + y².
c²+y².
y³x - 5c˚C j²x - 7:²D
дот де
D =
E=
8
6
Sic.
Here we might introduce all the variety of feries which express the areas of the conic
fections, circular fegments, &c. as alſo thoſe ſeries which exhibit the furfaces and
folidities of the ſphere, fpheroid, paraboloid, &c. but as theſe may be met with in
various authors, I fhall omit them. Several feries of this kind are given in Dr.
Hutton's excellent Treatife on Menfuration.
286. (F)
(E)
X
x
7 +2
ม− 2
1+z
n+ I
I
شير
x²+n+2
x- I
2m+n+2
+
+
+
m+1.m+n+ 2
•
2m+I 2m+n+2
1+2-2188
I
x
x
x
x3m+n+ 2
3m+1.3m+n+2
+2-3m
+ &c.
X:
+
+
+ &c. (x)
In
2+1
n+2-m n+2−2m n+2-3m
n+ I
Where
muſt be an affirm. integer.
m
12
n.ntr
n ntr.n+2r
+
+
+ &c.
m
m.m+r
m
m+r.m+er
n
m-
·1 -r
287. (F)
(=)
288. (F)
a+
a n
+
an.n-po
an.n+r. n+2r
+
+ Sic.
m m+r
m.
m+r.m+2r
•
*
m-n-r
a
(ε)
289. (F)
(S)
"
ада
m ± n + m± 2n + m± 3 n + m± 4n + &c. m±zn.
B
C
2.3
≈m" ± Arm"¬³n + 2 r. r−1.m²²²±r.r−1 . r — 2. m'—³µ³ +
2
·
D
2
r. r — 1 . r — 2 . r −3. m²−4n+ ± &c. till it terminates.
3. 4
A, B, C, &c. are the fums of the feries in Order III. Nos 6, 7, 8, &c.
250. (F)
a
c. İ
d-
e.
X:
+
b+c b+d
A +
B+
bte
-Ic+
C+
b+f
(I) Saxo
ax
b+1
6+ I
C 2
fD+ &c.
6+8
291. (F)
12
&c..
A SUPPLEMENT,
291. (F)
(E)
292. (F)
x
~
*
x
I
x
T
+
2+2
3**
x = L. x.
8.
4x+
+
10
fx
I
1
1
I
I
५
2 x
3 *
3
42
+ 5 ===
&c.
I
&c.-
1
(x)
Ꮮ
L. 1+
293. (F).
X
• + bx + c x² + dan³ text + &c.
x3
3nd A+2n-i.çВ + n −2.b C
3 a
5nƒA+4n-1.eВ + 3n-2.dC+2n−3.cD+n−4.6E
r
nb A
2nc A+n-1.bB
a+
x +
x² +
a
2 a
() {
4ne A+37−1.dB+2n−2.2C+n−3.6D
+
·x++
4 a
+
ба
5 a
6ng A+5n−1.ƒB+4n—2.eC+3n−3.dD+2n−4.cE+n−5.bF
**
+ &c...
A, B, C, D, &c. being the coefficients of the terms immediately preceding thoſe.
wherein they firft arife.
By means of this formula may the roots of any adfected equation be exhibited by
feries; and confequently when fuch roots can be had in finite terms, the fums of
thoſe infinite ſeries will be obtained. And from this formula will a great variety of:
infinite feries arife whofe fums will be had in finite expreffions by taking 2, 3, 4,
&c. terms of (F), and affuming different values for a, b, c, &c. and n. From
whence, by addition and fubtraction, we alfo derive a number of feries of a different
claſs, with their fummations; among which are exhibited the roots of certain adfected.
equations, of any dimenfion.
294. (F) 2ŵbx:1± +
I-n.c²
2 n² b²
ديد
+
1-I-2n.1-3n.c4 1~n. I−2n. I−3n . 1−4n . I−5# . c
2.3.4n+b+
2 3.4.5.6n b
6
+ &c..
n
(=) b+ √ ± c² +
';
2:
n
2
n.n-3.1
✔±²; being roots of the general equation
n.n~4.n−5 · r
3
x" In r
2
n.n−5.n—6.n—yor'
干
​2.3
4
1-8
= &c.
26=0.
2.3.4
Wherer (= b²) muſt be a pofitive quantity, or o..
}
295. (F):
•
A SUPPLEMENT,
13
&c.
2.6
295. (F).
X:
+
n
I-n. I ·288.b³
2.3n³c²
1−n. 1-2n.1-3n.1−4n.bª
+
+
2.3.4.5.n³ c+
C
(E)
I-n. I-
•
N. 1 — 2 n. 1—3n.1—4n
I-5n. 1-6n.ba
2.3.4.5.6. 7 n² c°
±c²+b
+ &c.
c²-b; being roots of the above equa-
tion, when the figns are pofitive, and r (¿² -¿') is a poſitive
quantity, or o; n being an odd number.
Hence arife the following feries,
2vbX:1-
3.5c
c²
2.462 2.4.6.85+
3.5.7.90°
2.4.6.8.10.126°
&c.
√b+c+ √b-c; being a root of the equat. x² — 2√¿² —— 21 —0.
296. (F)
(E).
297. (F)
24bx: 1-
2 c²
3.662
(E)
298. (F)
2.5.8c+
3.6.9.126*
b+c+ √6-c; being a root of the equat. x³ — 3√¿³—¿²
21/bx:131²
4.852
2.5.8.11.14c
3.6.9.12.15.186°
&c.
— 3√² 6² — c². x =¯2b..
3.7.11c.
4.8.12. 1664
3.7. 11. 15. 19c°
4.8.12.16.20.246°
&c.
(E) +b+c+b-c; being a root of the equat.x²-4b²-c²x²-2√ b²—¿²-2b=0..
299. (F)
bx
4c2
5.1063
(E) √/b+c+√/b= c;
4.9.14c+
4.9.14. 19. 24
&c.
5.10.15.206* 5.10.15.20.25.306°
b-c; a root of the equat.x-5√b²-c³.x³-5.6² —6²)³ x— 2b = 0.
300. (F) 24/6X:1-7.146
6.13.20c⭑ 6.13.20.27.36c6 &c.
7.14.21.2864
7.14.21.28.35·426°
2
23-
(E) √6+6+√c; a root of the equat. »
7
-7
•b³—c³ \ x³—7.b³—c²) x=b.
&c.
301. (F)
24e|bs.
I 2.562
X:
+
3
3.6.9c²
2.5.8.11b+
3.6.9.12.15. c*
+ &c.
(E)
c+b.
46-6.
26
302. (F)
4·9b²
X
+
+
5
5.10.15 c²
4 . 9 . 14 . 19 b*
5.10.15.20 . 25 c*
+ Sic.
(E).
√/c+6-√6-6.
&c.
く ​c-bo
"'!
If
14
&c.
A SUPPLEMENT,
If in the general formula N° 29.3. we take b =
303. (F) 1±
I
12
·
1 −n. I −2 n. 1 − 3 n.x +
+
2 3.4"
n+
2 n°
n
22) and c =
we obtain
2
6
1−n. 1—2n.1−3n.1−4n.1−5n. *° + &c.
2.3.4.5.6
3 • 4 • 5 • 6 n° :
2
11
(E)
플
​I +
√ ± x² +
I
I
And by making b =
,
and c
the formula N° 294. becomes
·H
2λ
2 x
2 x
I
I
Z
304. (F)
+
+
n
I-n. I 2n. 137. I
2 · 3 · 4 ·5725
400x4
±&co
•
(E)
I+
n . I — 2n.
2.3 n³
2 x
11
✓
I-
Hence ariſe the two following formulæ.
•
I. 2 3.4n+
1-n. 1-2n.t-zn.1-4n.x4
I-n. 1-2n. 1-3n.1-4n. 1-5n. 1-6n.1-7.x8
1.2.3.4.5.6.7.87°
n
#
"
I+x+
+
I -
305. (F) 1+
I-N. 1—2n. I—3n.x*
+
(E)
1306. (F) // +
+
12
1.2.3.4.5ns
"
1
n
I
2
(E)
X: I+x
2x
+
+&c.
I−n.1—2n.I−3n.1−4n.1−5n.1-6n.1-7n. 1-8n.x8
1.2.3.4.5.6.7.8.91
+&c.
See Dr. Hutton's Paper on Cubic Equations and Infinite Series, Phil. Tranf. Vol.
LXX. Art. 25.
d
307. (F)
b
I.2² d²
463
3
+
(x)
d
308. (F)
b
4.665
4.6.86'
The affirmative root of the equation x² + bx = d.
3
1.22d2 1.3.24 d³
I
I
1.3.24 d³ 1.3.5.25d4 1·3·5·7.2³ds
+
&c.
4.0.8.10by
b
463
4.6bs
1.3.5. 2° d^
4.6.867
&c.
(E)
The greater root of the equation — b x = - d.
2
X
-
309. (F)
2b X: 1-
1.2³d²
I
464
1.3.5.2°d+
4
1.3.5.7.9.21° d°
4.6.86°
&c.
4.6.8.10.12612
(E)
b2
b2
+d+N
d.
4
4
310. (F) 2b×:
d 1.3.2 d3
1.3.5.7.28d5
+
4.666
+
1.3.5.7.9.11.2¹2d7
4.6.8.10610
+
4.6.8.10. 12. 14614+ &c.
(E)
62
-+d
4
b2
d.
4
311. XF)
A SUPPLEMENT,
15:
&c.
I
b+c
I
b²±2bc+c²
311. (F)
Vax:1±
3
b³±3b³c+3bc²±c³
+
2
2 a
2.2
a²
2
2.3.23
a
3.5
2.3.4.2
b+ ± 4 b³ c + 6 b² c² ± 5bc³ +c4
± &c.
4
(E)
a+b+
MIN
I b±c 2
312. (F) VaX:1±
3
a
2.3
b²±2bc+c²
a²
+
2.5
2.3.3
b³±3b²c+3bc²±c³
&c..
(E)
a ± b± c
ຈ .
3
HM
&c.
313. (F)
m.m+n.x²a m+2n.m+3n.x²b
n. 2n
+
+
m+4n.m+5n.x²c
3n.4n
5 n.6n
+ &Tee
I
I
+
I-x
九
​(E)
I + x
n
Where a, b, c, &c. are the preceding terms.
By means of the following table of fluxions and fluents may a great variety of infi--
nite feries with their funmations be exhibited, by expanding the fluxionary expref--
fions, and taking the fluents. An example or two will explain it.
Form IV.
L
I
I
×, by divifion, &c. becomes
3.14. (F)
x
3
* + + +
+ &c. = (E) L
1 + x
5
7
1-3
J
X-
//
Form VI. f =
3.15. (F)
x+
+
5
9
+
13
13
I- *
+ &c. = (E) L.
Form XIX. f
2.
316. (F)
+
2
10
10
&c..
14
14.
I
X
x x x =
1 + x
+ 1A, t..
1
+ &c. = (ε) ¦ A, t . x².
{
And by the addition or fubtraction of fuch of thefe expreffions, the terms of whoſe
refulting feries may be fo connected that the denominators (or exponents) may form
one continued arith. progreffion, we derive a number of other feries, and their
fummations.
Lir
16
A SUPPLEMENT, &c.
317. (F)
X
f
S-S=
S
4
2 2
x
+ - - + 2 &c. = (2) √ } × L. 1 + z² x + 2²
3
5
7
Size + Sp=
x
✔
1 2²x+x
I
318. (F)
*+
اين
II
5
7
+ +
-+&c. = (E) { L.
1+x+x²
2
I + x ³
+ L.
II
3
I
x
&c.
From hence alfo may a variety of ſeries of a different claſs be obtained, by multiplying
the above ſeries and their respective fums by fome fluxionary expreffion, and taking
the fluents. And by repeating the operation, with the fame or fome other fluxionary
quantity, different feries and expreffions for their ſums will ariſe.
Six
319. (F)
X
2
+
3
+
&c. = (E) L. 1+*.
4
5
Multiplying this ſeries by x
*, and taking the fluent, we gt
320. (F)
x
+
2 32 42 2
+
&c. = (E)
(E)
X
fi
+x
321. (F)
Multiply again by x, and take the fluent, and there arifes
x3
12.2
22.3 32.4
+
I
This feries multiplied by x
.5
4+ 5+ &c. = (1) S & S + '& S
* * * * + ×
*, and the fluent taken, the reſult is
2
322. (F)
2
1.2
+
2.3 3.4
&c.
2
4.5
•
By the fame procefs we find
1.3 2.4
25
3.5
+
3
323. (F)
324.(F)
+
2
1.3.5 2.4.6
3.5.7
&c.
+ &c. (Σ)
x
*
(~) ƒ *¯' * S * S *¯* *S ï # *
+
+ &c. = (2)
ƒ »˜¯ ' * ƒ** £*¯* * Si
2
+
4.6
+
* ·
&c.= {~\ƒx¯'S* *S*~'S× S « Sür
(x))
326. (F)
1
A SUPPLEMENT, &c.
17
325. (F)
મ
326. (F)
2
&c.
² + - +
3
1
62
+
ΤΟ
7
x
14
10¹ 14
2
&c. = (x)
S
x
x
1+x
+ &c. = (ε) Sx− ' * S++x.
*
And thus may we inveſtigate as many ſeries of this clafs as we pleaſe with fluxionary
expreffions for their fums; the fluents of which may generally be exhibited by the
areas of tranfcendant hyperbolas, and in particular cafes by circular arcs.
this is the most trouble fome part of the bufinefs, we ſhall juft point out an eafy pro-
ceſs in the finding of the fluent of x
I
S
1+*
But as
by means of the circular arc, when
the ultimate value of x is fuppofed to be 1; by which procedure may the fums of va-
rious ſeries of this kind be diſcovered, and confequently the fluents of their corre-
fponding expreffions, when confidered under fuch limitations as arife from or apper-
tain to the circle.
The fluent of x
x
fit*
I + *
is expreffed by the area of the tranſcendant hyberbola
GMLC (feries p. 118.) the method of computing which (when x is 1) is there
exemplified. But the fluent of this expreffion may be more commodiously exhibited
for computation when x has the above value by the arc of a circle. For if the cofine
of the circular arc (a) be denoted by y, VI. a will be equal L. y+y-1 (fee
Simpſon's Mifcell. Tracts, p. 77.), and by N° 291. (feries) L. is =
I
+
2x²
3x3
*
&c. writing therefore y+V-1 for x in this feries, and re-
ducing the terms, &c. we have
{√ = 1. a = √ y² = 1-
2
2y Vy² = i
+
3y²✔/y³-1-y²-1]² _ 4y³√y`—1—4y•y°—1]”
+ &c.
2
3
4
a
Multiply the fides of this equation by
and
reſpectively, (the latter
I
quantity being the fluxion of the circular arc, kad. 1, cofine y,) and taking the cor-
rect fluent, a being evidently o when 91, we have
-
$
-y
a² = −y− 1 + 2 y² — 2 — y³ — 3 y . 1 — y² — 1
3ª — 6y³ . 1 — y² + 1 —y²)' — 1
−y
+
2
3*
4²
— ·
y³ — 10 y³ . 1 —y² + 5y •1—y³\'—1 + &c
エーアー
​D
Now
A SUPPLEMENT, &c.
Now take y (the cofine) = o, and confequently a = the quad antal arc (A), and
2
I
I
2
I
I
the refult is I
2
+ +
+ +
&c. =
2
63
9
1
I
I
I
I
I
I
-
I
I
+
+
2
2
2
3
4
5
+ = = = = 6
+
63
72
+
82
1/12 &c. = 1 A²;
I
2
2
which multiplied by
I
+
1
2
I
4
4
3
I
+
2
and the terms brought to order, becomes
&c. A', which is therefore equal to
3
I
X
in the circumftances above-mentioned.
I +
It is obfervable that the above literal feries y− 1 +
2x² - 2
22
&c. is equal the
−
y² − 1 + y²
+
2
2
y³ — 3 y . 1 — y²
3
2
I
I
&c. and i
+
2
2
3
-100
.&c.
y² - 1+ y²
22
+
y³ — 3 y • 1 — y³
32
&c.
-
A² = a². And (from
difference of the feries y
o,
that is, y
Mr. Emerfon's Trigon. prop. 26. p. 50.) it appears, that the numerators of the
terms of this feries are the refpective cofines of a, 2a, 3a, 4a, &c. (rad. 1); from
which circumftance Mr. Landen in his fummations of theſe kind of feries (Memoirs,
p. 68.) has introduced the fines and cofines of multiple arcs; from the well-known
values of which, when y (the cofine) is 0, +1, -1, &c. it more readily appears in
what cafes the numerators of the terms of the reſulting feries will become unity, hav-
ing the figns pofitive or alternately and; but it is very evident that this confi-
+
deration is not abſolutely neceſſary in the fummation, the fame conclufions being at-
tainable equally without it. The principal and perhaps only advantage arifing there-
from is, that we more eafily diſcover by this means than any other the ſum of the
feries when the fign of every other term is changed;, the cofines of the arcs
e, 2a, 3a, &c. being well known to be the cofines of the fupplemental arcs
2A—a, 2 × 2A-6, 3 × 2A—a, &c. having their figns and alternately, fo
long as the former arcs do not exceed a quadrant.
2
Now, from the above ſeries y — ³²=1+9² + &c. =
multiplying the fides of the equation by
fluents,
2
A² - 4a², we have, by
and a reſpectively, and taking the
3
2
2 y. Voz — y²
3D². v.
+
2
2
&c. = } A² a
23
3³
12
And
:
A
19
SUPPLEMENT, &c.
And if we take y (the cofine) = o, a will become A; and therefore
I
I
I
I
+
3
5³
7
3
+ &c. = 4A³. (See Series, p. 169)
and a as be-
Multiplying again the fides of the laft equation but one by
fore, and taking their fluents, there arifes
-y+
y²—1+y² _ y³ — 39 • 1—y²
24
But when y (the cofine) is
— y .
A2a2
+ &c. =
+
34
48
6
= 1,
a will be o;
hence the correct fluent is
-y-1+
2y22
24
y ³ — 3
7
. 1 — y² — 1
&c. =
3*
4
6
Now take yo, and confequently a
A² a²
A, and the refult is
2
I
I
7A+
I
+
+ — 2 − 23/4 + &c.
24 3 54 64
=
;
48
which multiplied by
16
and the
terms brought to order, as before, becomes
15'
I
I
I
I
7A+
I
+
2 3*
+
&c. =
5
45
&c.
By a ſimilar procedure with different fluents, as
S
袋
​&c. may
other feries be fummed, without the confideration of multiple arcs, or any peculiar
method of determining the value of the correction of the laſt fluent; but perhaps not
in ſo perfpicuous a manner. And from what we have before obſerved, that if , be
the cofine of the leaſt arc a, -y will be the cofine of the arc 2A-a, we may, by ſub-
ftituting theſe values for y and a in the above feries, derive a variety of others
with their fummations.
This method (with the affiftance of multiple arcs) is farther applied in the before-
mentioned treatiſe to feries whoſe denominators confift of the fame powers of two or
more factors in the fame arith. progreffion; but it becomes more perplexed and em-
barraffed, and more liable to exceptions. One principal objection is, that when the
feries is brought to the propofed form by continually multiplying by the fame or
fome other fluxionary expreffion, and taking the correct fluents, the laft correction
of the infinite (or feries) part of the fluent must be had in known terms before the
general expreffion can be exhibited; unleſs that correction be exterminated in " bring-
ing the two feries together in order to form only one feries," but in this caſe you do
not obtain from the general expreffion the fum of that ſeries into which you had ulti-
mately refolved the equation by taking the fluent. Inveſtigations of the general
D 2
theorems,
f.
F-
20
&c.
A
SUPPLEMENT,
thcorems, Art. 11, 13, (Memoirs, p. 74, &c.) will fufficiently explain this ab-
fervation.
Let the two feries which exprefs the value of L. 1+ (No 292, 319,) be equated,
and fo adjuſted as to vanish when x is 1; multiply the equation by x, and take
the correct fluent; multiply again the refult by
and correct the Auent; fubtract one
x
fide of the equation from the other, and multiply the remainder by
and there arifes by tranſpoſition
I
I
x:
2 √ — I
X
3
I. 2
7
2
x
x
+
&c. =
22.32
32.42
2 ↓ = × S = S & S & S ÷ +2px−2p+K+K.X−2ÏX − {x²; where
21
X
X
X
P, R, R are the reſpective corrections of the feries part of the fucceffive fluents, the
fourth correction S being exterminated in the fubtraction of the expreffions; and
X
= L. x.
X
x = ƒ==
√
Now for SSSS write its equal
=
X
Form L.); alfo for X write V1.a; for p,
I.
the latter fide of the equation reduces to
2
X2 x
2 X x−X+3×−3* (by
A2
; and for Ŕ +Ŕ,
2 A 2
I, and
3
3
|
x
X 2
1.ax+21-1.x
2
+ *¯¹×e. Laſtly, for x write y+V-1,
xa.
x
2 A2x
2A2
2
a x
a²x
×3x-3+
+
2√
I
3
3
2
-플
​x
2A2
a
X
142
+ 3-
*
3
2
2 √ - I
and we have
MIN
2.
in't
5.
-y-
2
-y-
-
2
I
I. 22
+ &c.
1.22
22.32
¿A -
2
3
2
+3−y+Vj³−1} + y −√y²-1)*x a,
X
y + √ y ²
2
2 √ - 1
The theorem in Mr. Landen's Memoirs, p. 138. which fhould exhibit the fluents of thefe compound flux-
ional expreffions, is erroneous in every term after the fecond.
This theorem, however, may be found truly expreffed in Simpſon's Fluxions, p. 390. which is effentially the
fame with Form L. in the following table.
which
A
22
SUPPLEMENT,
&c.
which is evidently Mr. Landen's general theorem, p. 75.; an expreffion that does not
feem reducible by any means to the numerators being unity and the figns alternately
+ and
which is the form of the feries part of the laft fluent when the ultimate
S
x
value of x is 1, and to which it is well known the expreffion + S & S S
Si++
is equal, fuppofing the fluents to be generated while x from o becomes 1.
The rea-
fon of this is indeed evident; for fince the last correction (S) is exterminated by tak-
ing one expreffion from the other, the feries is therefore entirely deſtroyed thereby in
the equation, and confequently cannot be deduced therefrom by taking the variable
parts of the equation o. We may however derive the feries when the figns are all
affirmative, by ſubſtituting 2A-e for a in the above equation, which will then be
transformed to
ཏྭཱ
Σ
x + √ y²
Enday
$
+ &c.
11
3
3 + √ y²
2
2.1.22
+y-
2.2². 3²
Me
114
J
-y-
2√
[melet
× 4A ~ 20 + 3 + √ y².
X
+y-
2A-
-A a +
3
| +
1
3
2
i
X
2
in which taking y (the cufine) = 1, and confeq. a = o, we have
I
I.
+
2.3*
I
4 A 2
+ &c. =
-
+
1-2 3-4
22
•
In Art. 13. (Memoirs) by proceeding from the two values of
292.) as the latter fide of the equation denotes, we have
2.4
P
f
+358 + S ÷ - Ŕ ƒ ÷
+
&c. +
2
2
4
&c..
4
2
x
S=S» SS + − S
IV
R
4.
I
S=
+ £ - 5 =
4
S
x
+1+x.
S
I+ &
در
દ
3:
(Nos 320,
px
4
SSS S
+ *
S
- 4 - St S ÷ + S t + 8
*
2
لة
x
x
From whence, by fubftituting the values of theſe fluentials (Form L.), fubtracting,
dividing by x, and tranfpofing, we have
+3
32
x
1.3
inc
x²+x
r. 22
x³ + x
22.4
3
**+x-4
+
&c.
32.52
8
+
I
8x
X²
"
180
1V
K R
7 X
1
+
+
X+
X
* 8x 4
2
2x 16
+
2/03
3*
13
+
− 1 + 2 Š x ;
16x
wh re
22
&c.
A SUPPLEMENT,
IV
where p, K, R, and S, denote the refpective corrections of the fucceeding fluents,
and X as before. Now from hence let us endeavour to deduce Mr. L's general theo-
rem (p. 77.) by fubftituting according to his method,
IV
and 2 S for K-R, and we get
A2
"
I.a for X;
for pi
3
A²x A2 13 3x
+ +
I
A¹a
+
6 6x 16x 16 2 3* √ ~I
7 a
8x√-I
2 Sa
xa
a²x
+
P
4V-I 8
1
a
2's
+
8x x
Here then it is evident that Ŝ is not exterininated, therefore its value muft neceſſarily
be known before we can deduce the theorem adverted to; accordingly writing
A2
for S, we have
6
16
да
2 Sa
x √ - I
2
2
A² a
3x1-1 8x-1
and therefore by fubftituting we get
x²+x
I.
2
૨૨
x³+x-3
a
2 S
and
A² 13
+
A ²
2
4
x 6x 16x
: +
6x
+ 3x
16.x ›
X.100 X
A2
2
+&c. =
+x+ㄨ​ˇˋ
xx X
+
4"
6
8
3
16
I
1
|
; or,
2
1 — y
by writing y + 1 for x, and dividing the whole by 2,
'y² — a,
j² — 1+ y² — y³ — 3 y ·
y² — 6 y². 1 − y² + 1-y²]²
—
2
2
2
+
&c. =
I.2
2 4
32.5°
A2
3
I
I-y² x
+ y x
+
; which is obviously Mr. L's theorem,
8
16
2
4
P. 77.
It is therefore very evident that Mr. Landen in the inveſtigations of thefe funda-
mental theorems has been beholden to ſome procefs that he has not premiſed (at leaſt
I
in theſe Memoirs), the ſum of the feries
12 33
I
2
.4
+ &c. being requifite
to be known previous to obtaining the theorem, p. 77. and that of
I
1² . 3² • 5²
I
I
I
By No 130. we have
12. 38 + 32.52 + 52.72
A2
+ &c. =
4
I
I
and
23. 42+ 43, 62+ &c-=
I
I
X:
I
I
16 × : 1722 + 23, 32+ &c.
I
I
12
A2
3
; and therefore by addition and fubtraction, we have
16
I
A 2
II
1
I
I
Az
2
·
x², 32 + 23.42 +
32.52
+ &c. =
1
and
3
16
1. 37
2
+
5
&c. =
32.52
6
16
I
2 .4
42.6 ²
A
A SUPPLEMENT,
23
&c.
i
T
I
2².42.62
+
3*·5¹·7'
&c. previous to raifing the theorem, p. 79, and this muſt
neceffarily be always the cafe when the laſt correction (3) is not exterminated; and
yet theſe very ſeries are the examples exhibited as the refults of thofe general the
orems, when-x is taken 1,! Is not this what a Logician would call a Fallacy,-.
A Begging of the Queſtion? Perlraps Mr. L. may think proper to ſet this matter in a
clearer light in foine future publication; as it rather appears, that the fums of the
above particular ſeries ſhould be always the neceſſary poſtulata for obtaining thoſe ge-
neral theorems; and confequently that the method of fumming them ought to have.
been previoufly confidered.
By means of the Forms LIX, LX, &c. in the following table, will the fummations
of a different order of feries arife, viz.
327. (F) x- +
3
I
x³ +.
3.5
+.
3
I
2.4 3.4
2.4.6 3.2.6 5.6
+
3.5.7
3.5
2.4.0.8 3.2.4.8 5.2.8
3
[
+
7.8
¹+_3·5·7·9
+ *
+
+
·
3.5.7
•*- &c.
2.4.6.8.10 3.2.4.6.10 5.2.4.10 7.2.10 9.10
3.5
3
Ι
+
+
(E)
I
I
a+
+
x
x
I
1+x²
328. (F) *
3
I
+
2.4 34
5 3.5
+
+ &c. x9.
2.4.6
3.5.7
2.4.6.8
13
+ &c. *¹³ + &c.
(E)
X
at
-1
I
I
I
I+.
I + x +
3
329. (F) *- 3 +
3.5
+ &c. x I 3
2.4 3.4
2.4.6
(E)
&c.
*
+
19
&c.
3-5-7
2.4.6.8
+ &c. x¹9
I
a +
5
I + x²
Where the law of continuation is manifeft; a being the circ. arc, rad. 1. tang. re-
ſpectively, x, x², x³, &c.
From hence may a number of very difficult feries be fummed by taking different.
values of a. For instance, let a A (A being the quadrantal arc, rad. 1.) and
* will be
I
√3
1
330.(F).
; which values being fubftituted in N° 327, there arifes
3 I
+ X +
2 32 2.4 3.4
+
I 3.5
3
I
I
35
2.4.6 3.2.65.6
+5.6x
=(x)
- &c, = (£) ~ 3 • ^ - √ 3 + 3.
3.A
3 ₹
2
Aga'n,
24
, `&c.
A SUPPLEMENTMENT,
Į
!
331. (F) 1×3
3
2
+
Again, let a A, and by N° 328, we have x = 34, and therefore
1
3.4
34,
3.2.6
+- × 34 + 3.5
3++
3
I
+
+
X × 3 =
3
&c.
2.4.6
5.6
I
I
(E)
A
3
3 +
2034
Let a again be = A, then will x =
टु
I
3
H/M
τ
in N° 329, hence
I
332.(F)×
Him
Τ
7
√10+2√5
3
3
+
2.4
1
3
'་
+
3.4 10+2√5
3.5
2.4.6
+ &c.x
√5-1
√10+2√5
3
この
​(E)
10+2518. A 10+25-4.10+218
+
2016-2√5
4.√5-13
उ
A
If a be
A, x will be 1, and the fum of each feries =
1- √2
+
212
√2
x3x
25%
+
3.1 + x
I
5 • 1 + x
The above theorems are thus inveftigated. By Form IX. (in the following table)
we have
Six
+5
7
x
= X
+
I + x
3 5
7
multiplied by
xx
2
1+x²
terms are reſpectively
+ &c. (a) circ. arc, r. I, t. x, which
(the fluxion
the fluxion of the fine
becomes
I+x
ax
&c. =
Now, the fluents of theſe
2
I + x
2
S
xx
3
et is le
11
32-4
+
3·525
6
&c.
X
+
3
I
5
X
S.
2-3 x
+ *
x5%
I+x
2
ረ
2
Ι
2.4
2.4.6
11
11
1+
HIM
3
4
+
2.6
3x6
&c.
I
X
5
+
&c.
6
&c.
Therefore
3
I
+
to
× *4 + &c.
2
2.4
3.4
=S₂
ax
x
*-
is (by Form LX.)
theorems inveſted by means of the fluents
a+
I.
2
I+x
xx
theorem will refult from the fluent
12
S
I
2
X
20
I++"
; the correct fluent of which
I + **
3
In like manner are the other
J
&c. and a general
I + x
1
A TA-
1
&cc
A SUPPLEMENT, &c.
25
A TABLE OF FLUXIONS,
WITH THEIR FLUENTS CALCULATED:
Intended to facilitate the Computations of the Sums of the Series from
N° 262. to 269. inclufive, when they are attainable only by Means of
Circular Arcs and Logarithms; the Formulæ, in fuch Cafe, being, ir
general, ultimately refolvable into thefe Fluxional Expreffions, and
Algebraic Quantities.
L, denotes the hyperbolic logarithm; A, the circular arc, rad. I; s, the fine;
t, the tangent; a = √ √ 10+2√5,
√10-2√5, c = 1 × √5-1
d = 4 × √5+1.
b
=
I.
1. S**
12
x
- - L. I
L.1-XX
I
n
x x
II.
I + x
2
23
2
3
+1
±L.1+* = *
1+x=x ±
10/
४
K
دی
در
X
&c.
77
1
+
Sic. +
4
n
Where the higher or lower fign obtains according as ʼn is even or odd,
#- I
X
X
I
III.
L. IF*".
n
IF x^
IV.
S
x
I + x
= {L.
I
V.
VI.
VII.
S
S
S
I
I
I
VIII.
I -
x
= ÷ × : { L. 1+x+x²¬L.i−x+ √ 3. A, s.
||
+
I +V
x : L.
+ 2A, I. N.
√3.x
2 √ I + + *`
3 :
I
I
x : — L.1−x-cL. 1 − 2 c x + x−dL.1 +2 4 x + a
+2a A, s
L.
X: L
a x
+ 26 A, s.
b x
1-2c+心
​1 + 2 d x + x
+ L.
I
+ √ 3. A, t.
√3.x
པ�ས ང;
E
I + x + x²
(
IX.
26
A SUPPLEMENT, &c.
2
= A,
+ x
A, t. x.
11
11
} × : L. 1+x - { L. 1−x+*² + √ 3. A, s
√3.x
2V
I-x+
IX: L.
I+ √2.x + x
1−√2.x+x
√2.x
+ √2. A, t.
I
2
{X: L.π+x-d L. 1-2 dx+x²-c L. 1+2 cx+x²
I
I + xs
+26A, s.
b x
√1-2dx+x
+2aA, s.
110
X:
• √ & . L. 1 ~ √ 3 • *+*
2
a x
√1+2cx+x.
I + 2 cx + w
√3.L.1+√3 • * + x²
x
+ A, s
+2A,t.x+A,s.
21
1- √3. +
2
*
I
X
2 √ 1+ √3 • * + *·²
IX. Sit
X.
XI.
x1. S
XII. J
Si
I + x
I + x
XIII. S₁+x
XIV.
XV.
S
xv./
XVI.
I
1
—
-it's
3
= // × : / L. 1+x+x²-L. 1− x − √ 3. A, s.
× L.
1 + x²
I
--
*-
I-
༧/༢ .
2 √1 + x + x
2
x :d L. 1-2 cx+x²-cL.1+2 dx+x²-L.i
X
S
x x
ах
br
J
I
*
+2b A, s.
-2aA, s.
√1-2cx+λ²
1+2 dx+x
x x
I
4
X
{ x = { L. 1 + x² +x*~L. 1 − x² + √3. A, s
√3.22
2√
2
I
XVII
× {
XVII.
f = X = X: L. 1−x+x − L. 1 + x + √3 . A, s.
x x
1 + x³
24/
√3.x
1 − x + x'
XIX. SIXAV
x x
1A, t. x².
+ x
× :cL.1+2ax+x −dL.1—26x+x² - L. 1+*
XX. Si
хх
b x
a x
S
=
+2b. A, s.
2a. A, s.
√ 1-2cx+x²
XXI.
"
A SUPPLEMENT,
27
&c,
2
2
† × : L. 1 + x² — { L. 1−x²+x*+ √ 3. A‚· 5.
√3.x²
2√1
I − x + x+
XXI. Si
XXII.
XXIII.
S:
√ ₁
хх
xx
I x
= 1 × L.
Sxx =
I - X
6
X :
L.
I
3
I+x
I X
2 A, t. x.
V플
​Z
1 − 2² x + x
2
I- 2
X
XXIV. *** =
1 + x
XV. L.
I+2=x+x
+ √2. A,
S
2
-
+√2.A, 5.
1+2 = x
XXV. S²***
x+x
•
x x
1 +1.0
= A, t.x³.
x3x
XXVI.
= ↓ L . i +x¯‍+x²− ¿ L. 1 — — 2√3^, .
I
6
I
A, t.
√3.x²
XXVII.
S
S
XXVIII.
XXIX.
S
S
I
x3x
I+ *
I
xx
I+
6
= ÷ × : { L. T−x+x²−L• i+x+ √3. A, 5.
s.
2+x
༧ ༣
1 x+x²
13.x
× L.
I + x + x²
1−x+x
+ L.
- √ 3. A, t.
I-
x
x: 3A, t
I - X
2 A, t. x³ — {√3. L
1+ √3.x++²
-
X
The following forms are fuch as ariſe in Orders 27, 28, 29, 30, when the ferie:
are not reducible into pure algebraic expreffions; and as they are here generally ex-
preffed, fome of them will require ad ufting to the propofed feries, by obſerving the
refults of both when x is fuppoſed to vanish.
XXX.
XXXI.
S
S
&
1 .\
४
Hla
I x
4/1
[r]
=
= A,
A, s. √³, √x-x².
XAXII.
E 2
28
&c.
A SUPPLEMENT,
XXXII.
S
XXXIII
1
I
x x
3
x-x
XXXIV.
I
THIO
Hier
11
— ± √ 1 − x — § √ x − x².
3
2
= − {x?√T−x− &√x−x² + 4A, s. √x.
-10
ㅎ
​४
+ 2 = x² + 18 x + x
1−x + î¿Â, s • √/ x.
5
16
2
=
I
x
XXXV.
S
XXXVI. S
XXXVII.
I- A
I-
H/2
3
xx
=
Ι
XXXVIII.
XXXIX.
S
XL. S
XLI.
XLII.
XLIII.
S
S
I
Ι
Ho
R
I + x
x x
I + A
5
HIN
2 x
√ I-X
4- 2 x
3:
I
X
3
X
√ 1 - X
7
2x2
=
I
2A, s. vx.
·
3A, s. vx.
15
+ 2 ׳ + ¿x² + ¦ aª × √ π—— 1¿A, s. √x.
I x
= 2√I+~.
=x+x
-L. *³+
I+x.
2
x² + x³ — ± √ 1+x.
² + √ I + x + √ x + x² + {x ? . ✓ 1 + x•
L. +
I + x
S
2
11
L.
² + √ 1 + x + x²+
KLIV.
XLV.
S
S
X
1 + x
3
2
I + N
M-1
3
X
|
2
+
५
via
X
1+x.
3
Iso
XLVI.
A SUPPLEMENT,
29
&c.
XLVI.
XLVII.
XLVIII.
XLIX.
S
S
S
*
2
1 + *
+
1 + *
1 +
•
= 2 L. x² + √I+x+ √x + x² +
-
4+2
·3 L. x²+ √i+x+√+ +
3 A
K
+
3
十七
​$ L.
V
I+x+x+x`* +
3x3
এই
I+:
+
21
x
1
12
x
771
S
S
X
X
I
1- I
+
N.N-1
3
I
x x
+ &c.
N. x— I . 11 — 2
z x.
m³
L. mx
-m
·S:
M-I
x
S=
I
LI.
S & S x =
x S5 - Sy
LII.
S S y
ż
- Sz
LIII.
Szu. Sy
z
ƒ * S * ƒ ÿ ÿ = ƒ ãù · ƒïãï − ƒ ƒSää.ÿâ+ų fyzi− Suyz¿.
Where the relation of u, z, y to x are given.
SS
W.IV. S & S //
x
X
LV. SSS
x
| ×
X
2
น
= { X² — { C², = 1 L..
= XIXX-C-TX c
x
C
Z
X² — C² — 2X +2X~+C− 1 X c.
2
x
ż.
= X² — C² − 4X + 6 × − +U−1.X+3C-C³~3X c.
LVI.
S = S · S✩ S = x²
2
LVII. S**SS
**
X
2
+C-
4
+C={.X-C²+
LVIII.S Sx+S+ S = X - C² = 2X + ? . ~ + C-4. X-C² + IC-4. —
X
&c.
*
Where X is the hyp. log. of x, or fuent of
Aluent is, and C the hyp. log. of c.
x
8
4
; c the value of when the
If c be taken = 1, theſe theorems will be adjuſted to the ſeries in p. 20, &c.
£ 3
LIX.
૩૦
A SUPPLEMENT, &c.
LIX.
2
-- I
xa”+n√ I—x².a” — 1 — n.n— 1.x.a
[ q”~1
Säz
ах
+n.n-I.n-2.n-3.x.a
11 — 2.-
n
-4+12··
n.n—I.n—2.V
2
n
3
I
x .a
2.
n
n 4. I
a
. &c.
denoting the circular arc, of which the fine is x, and radius 1.
a i
=ƒ
t i
2
I+1²)
3
LX. S
LXI.
S
at i
1 +14) 2
=Г
I+
I + i
=
||
Σ
t³ i.
3
+S..
is i
5 • 1 + 1²]
12
LXII.
S=
a t² i
&c.
3
S
£5 i
I+1
3
£2
t
-S
M
3.1+
a +
ť i
3.1+ +14):
3√ 1+10
a +
2
II
3.1+1°
I
a +
3
3
I
+
S
I
+S
I
II
i.
5.1+14
1+14
5.1+t
1+1°
M'N
I
2
Lanky
I
3
&c...
&c.
&c.
Where a is the circ. arc, rad. 1, and tang. refpectively, t, 12, 13, &c.
Generally,
Σ
2
LXIII. S
3
S
3-
S
12n-1 ¿
+
n
3.1+
f
tз" - I t
&c.
5 • 1+1
2
12
a
I
+
I
n
It
42
72
I
n
1 n
n
2
a the circ. arc, rad. 1, tang. t and n any even pofitive number.
This theorem holds good alſo when n is any pofitive number whatever.
EXAM-
A SUPPLEMENT,
3r
&c.
EXAMPLES TO SOME OF THE PRECEDING THEOREMS.
1. Let the feries propofed be
I
5.6.7.15
I
I
+
6.7.9.17 7.8.11.19
+
+ &c.
=
=
4,
This feries belongs to N° 265. from whence we have by compariſon
'q = n = 1, m = r = 5, s = u = 2, t = 13, and ·.· ″ = {, 1, and w = 4i
which correſponds with the first particular cafe, w and —♪ being whole numbers.
Hence by fubftitution we have
-
(E)
I
I
I
X
+
+
I
2.4.-.- 25+2 5+45+6
the fun of the propofed infinite feries.
I
I
+
5+8
+ 2.2. . - 1
5
こ
​X
I
49T
=
;
5 675075
By comparing the fame feries with N° 272. we find h = 5, e = 1, g = 3; and
thence by fubftitution derive the fame conclufion.
I
I
I
+
&c.
11.13.16.32
2. Where the feries propofed is
9.11.12.28 10.12.14.30
Here we find p = 8, q = n = 1, m = r = 10, § = u =
T = 5, d = 3, w = 8, where w and ≈ are even numbers.
theorem we have
2, t = 26, and ·.·
Hence by the fame
I
I
I
I
I
();
X
+
+
2.8.-3.-5
12 14 16
18
-18
I
I
I
20
-18
I
I
I
+
|
+
X
22 24 26 2.2.5.-3.2 9
IO
631
34594560
This ſeries being compared with N° 272. we have h = 12, g = 6, e = 2; and the
fum as above.
I
I
I
+
+
+ &c.
3. The feries propofed being
This feries is found to correſpond with the fecond particular cafe of the above
thcorem,, and ♪ being refpe&ively to 4, and . From whence by fubfti-
1.9 X 3.5 2.11X5.6 3.13X7.7
tution we have
I
I
I
I
I
1
(1)-
2.2.4. ½/ • ½
XI+
+
+
+
X
7
7
2 3 4 2.1.-3 3
| -
+
I
7413
+
5
グ
​740880
Comparing the fame feries with N° 273. we have g = 1, e = 3, h = 2; and
thence deduce the fame conclufion..
4. The
32
A SUPPLEMENT, &c.
4. The feries to be fummed being
τ
3.4.6.7
Divide this feries by 4, and it becomes
I
I
I
+
&c.
I
4.5.7.8 5.6.8.9
+ &c.
3.8.12.7 4. 10. 14. 8
which being compared with the fame formula (265.) we have p = 2, q = 1, m = 6,
But here - (-2) is negative, there-
fore according to Remark 2. N° 273. write for w, and w for ; that is, write
m, r, n, s, for r, m, s, n, reſpectively; and by ſubſtitution we ſhall have
&c. from whence w = 1,
d = 39
T = 4.
I
I
I
(2)
X
2.
.2.4.3
3
4
+
1
5
I.
6
+
I
which, being multiplied by 4, produces 720
>
I
w
2. 3. — 2
I
I
I
X
100
ΙΟ 2880
the fum of the propofed feries.
In comparing this feries with N° 273. we have (w-d) e negative, writing there-
fore ♪ for w, and w for d; that is, taking r = 2b+g−2e+2, and m = 2h+8+2,
we get e = 2, g = 2, and h = 3; and the refult the fame as above. And thus
may the fum of any other feries be found, provided they come under the limitations
ſpecified, when the general formulæ ſeem to fail.
5. Let the ferics propofid be
3.8x5
3.6.12.15
+
5. 13x6
6.8.15.18
+
7.18x7
9.10.18.24
+&c.
This feries correfponds with N° 269. or N° 270. Comparing the terms with the
latter, we have a = 1, b = n = 2, c = q = s = u = 3, e = 5, p = 0, m = 4,
-2, P
2, I,
k = 2,1 = 3,
2,1=3,
"
r = 9, t = 12.
From whence a = 1, B = 3; ά:
3
7
α =
= }; α =
a
+1
B =
दु
I 2 ;
a
-?,
?, B =
1/7 ; # = 2,, P = 3, 。 = 4; π
-
"
#
ó
=
σ = 2; π =
·3, P − 1, " = 1; * = −4, P = -2,0
σ
T
- I; k
20
t
be
5
I
and therefore
4 ;
;
hence A =
B
•
ย
q ns u
27
432
10
C =
D=
27
119
434
7
12
72"
Thefe values being written in the formula (E), and X (the ultimate value of x)
taken = 1, the refult will be
7
3 X :
72
+ 3/2 + 6
I
IO
I
X
27
+승
​J1Q
1
I
I
I
+ + X + + + =
3 6 9 4.32 3 6 9 I 2
the fum of the propofed infinite feries when the ultimate value of x is unity,
21 I
5184
6. Re-
A SUPPLEMENT,
33
&c.
33
6. Required the fum of the feries
5
6.3
+
1.2 2.3.4
7.9
8.27
+
+
+ 3:c.
Here we find a = 1, b = 1, p = 0, 9 q = m
m
is =
1, and it appears that x is = 4.
n
3.4.16 4.5.64
n = 1; and becauſe is 1,
Therefore in order to adapt this ſe-
ries to the general one, ſuppoſe each term to be multiplied by x², and by fubftitution
× −
-1; and ·.· A = 4, B = −3; hence
L.1-*+*² × 3, and taking * = ½, and dividing by
x
we have
www.
α 4, a = 3, x
3, x = 1,
(E) = × × 4 - 3
× -
X
2
(x²) the reſult
7. Let the feries propofed be
19 x³ 28x+ 39x5
is 4, the fum of the propofed feries.
+
+
52.45
67* 84x
+
+
1.2.3 2.3.4 3.4 5 4.5.6 ' 5.6.7
5.6.7 ↑ 6.7.8
This feries is evidently reſolvable into theſe two
+
+ &c.
19 x³
28*4
•
I 2.3 2.3.4
+
+
37x5
3.4.5
+
46 x6
557
4.5.6 5.6.7
+
+ &c.
+
+
+
+ &c.
I . 2 x³ 2 3 X 3.4x7 4.5
3.4.5 4.5.6 5.6.7 6.7.8
By comparing the former we have a = 10, b = 9, p = 0, q = m = n =
Į
W
r =.2, 1 = 2; and therefore
s
5 = 1,
=2. Hence - = 1,
1, p = 2,
A = 5,
I
4
X
*
I
1
X
B = −1, C = −4; and by ſubſtitution (₹) = ת × 5 -
+ *³ × = + — 2 × 1
x
x
2
+ 2/2 = 5**-*−4 × −L. 1−x+3*²+4×, the fum. And
2
in the latter we find a = 0, b
w
T = 1, p = 2, /= 2;
S 1, p = 2, m = 3, r = 4,
= 4, and thence A = 1, B = −6, C = 6;
6 6
Х
x
2 6
and by ſubſtitution (8) = x* × : ---- + × √ * * + * ײ = 3 × ÷ +
x
X
I
3 4
= (becauſe *** is = −L.1-x-x-, by Form 1.) x²-6x+6×−L.1−x
(becauſes
I
2
+3x²-6x, the fum. Hence the fum of the whole feries will be
? — 7x+6x²³×-L. 1−x+6 x² - 2 ×x; which will evidently be algebraical if x be
F
taken
34
taken
tion are
=
A. SUPP. LE ME. N T,
EM EN &c,
a rcot of the equation 6 x² - 7 x + 2 = 0. Now the two roots of this equa-
and; the former of which being written for a gives
19.23
1.2.3.33
+
28.24
39.25
2.3.4.3* 3.4.5.35
+
+ &c. = 1},
and the latter
19
3. 23
+
28
2.3.4.2
+
39
3.4.5.2³
+ &c = 1.
1. 2. 3.
S. Let the propofed feries be
Circ. are, S.
√8-√3
2.3
+- arc, s.
√15-√8
3.4
+ arc, s.
√24-√15
4.5
+ &c..
This feries is evidently equal to
9
- arc, s. § √ 15 - $ √ $+ &c. arc, 9. √ √ I-
}√2 + 1 75 \√÷ = s. §√ī−÷—¡√
arc, s.
+ aic, s. VI 16
I
VI + &c.
उ
Hence, by comparing this with
N° 274. we have a = 1, b, c, &c. decreafing quantities; and r = 1.
Therefore () = Circ. arc, fine, rad. 1. = 52359& &c. which agrees with:
N° 239.
•
9. Let the feries required to be fummed be
Circ. arc,
t.
5000
9950
+ arc, t
5000
9851
+ arc, t..
5000
9753
+ &c..
This feries appears to be equal to
Circ. arc, t.
10000
10000
10000
+arc, t.
+arc, t.
+ &ic.
100²+100X99
1002+99×9 &
100²+98×97
which therefore correſponds with Nº 276. from whence we find a = r = 100; a, b、
c, &c. decreaſing quantities; and therefore () = Circ. arc, tang. 100, rad. 100 =
78.539 &c. is the fum of the propofed feries.
The theorems to which theſe examples are given, are ïnveſtigated upon this moſt
obvious principle, that if there be a feries of quantities a, b, c, d, &c. any how de-
creafing till they vanifh, we fhall have a-bb-c + 6-d te d-e &c. a-e=
(when the laſt quantity e is = 0) 2. Whence, if a, b, c, &c. denote the fines of
circular-arcs, we obtain the firſt theorem;. if cofines, the fecond theorem arifes, &c.
10. Required the fum of the infinite feries
4.32
I
2
5.10.5°
4·9 · 14 · 3ª
5.10.15.20.5
4.9.14.19.24.3°
&c.
5. 10. 15. 20. 25 · 3Q .5
Comparing this feries with N° 294. we get b = 1, and . = 2; hence, by ſubſti-
tution,
A SUPPLEMENT,
35
&c.
tution, and reducing, there arifes
(x) =
√ T + 1 + √ T − 1 ; √/ 2+ being a root of the equation
*³ — 5׳ — 5» = {.
5
21/
11. Where the feries to be ſummed is
I
6.42
7.14.5
6.13.20.4
7.14.21.28.5*
√ į
6.13.20.27.35.4°
7.14.21.28· 35 · 42 · 5′
&c.
By comparing this feries with the fame formula, we find b = §, c = 1; and
therefore by fubftitution, &c. there arifes
=(3)
7
F
√ I + I + √ √ = 1; √/ 3 + √ being a root of the equation
* 3 - 7 *
7
2 J
{
12. Required the fum of the infinite feries
1/2-1/3
༧༢ 3
2
+ X
2+√3 2.4 3.4.
26-15√3 3.5
26 +15√3 + 2.4.6 3.2.0
3
+
x
362–209√3
5.6
362+209√3
3.5.7
3.5
3
I
2.4.6.8 3.2.4.8
+
+
+
5.2.8 7.8
5042 −2911 √3
X
+ &c.
5042+2911√3.
Comparing this feries with No 327. we find x =
2 - √ 3
which is evidently
-
the tangent of the arc of 15° radius 1. Hence a
have
(E)
11
√2+ √3
A +
12
2 ± √3
>
A; and by fubftitution we
2+ √3-2√2+ √3¸,
2√2-√3
the fum of the propoſed ſeries.
:
*
:
:
EMEND A-
F 2
1
36
A SUPPLEMENT, &c.
EMENDATIONS AND ADDITIONS.
4
Some Typographical Errors, and other Inaccuracies, having inadvertently
efcaped both in the Original and Tranſlation of Mr. LORGNA's Treatiſe,
it is prefumed the following Remarks, &c. will be of Service to the
Reader.
IN the enunciation of Prop. I. p. xi. for Number of Terms read Number or Place of
the Term. For the former part of the remark in p. 10. ſubſtitute this: Since the flu-
ential expreffion
P
x
I
9 ÷
is ſuppoſed to be generated while x from o becomes 1,
9
IF X
*
it is evident that the feries, and confequently the expreffion for its fum, ought, when
x is under the former circumftance, to be alfo = 0. It is therefore to be understood,
not only here, but in every propofition, that each fluential expreffion, when reduced,
is to be corrected by taking x = o, and ſubtracting the refult: And by finally ſubſti-
tuting 1 for x, that the expreffion is properly adjuſted to the propofed feries, which
is then faid to be perfectly integral, that being fuppofed its whole or ultimate value;
which is indeed actually fo when the figns of the propoſed ſeries are poſitive, as that
value of x manifeftly exhibits the whole fluent of the expreffion when the higher fign
takes place in the denominator.
P. 16. 1. 9. for the femicolon write C. 1. 5. from b. r.. - Sy z xe.
1+y.
P. 22. l. 16. r. an even number. P. 24. 1. 2. from b. r. L. 1+ y.
P. 25. l. 15. for + L. 1 − √ ≈, r. + L.—√x. l. 16, for L ..2. r. L. f. l. 17.
2
for L.
I. L.
"
√3°
4
3√3
P. 28. 1. 2. draw a vinculum from the first fluential fign over the whole expreffion.
1. 8. for latter fluential becomes, г. fluentials become.
X
π
P. 29. 1. ult. for r. P. 38. 1. 2. r. B = +1; 1. 8. for −1, r. + 1};
1.8.
1. 11. and 12. for x, r. y; 1. 19. for root r. quantity; 1. 26. for √ 3 • { √ 3 • Y, г.
√3.√3.1. P. 44. 1. 19. r.
I
3.4.6
P. 46. 1. 3. for + &c. r. - &c. P. 47.
1. 4.
A SUPPLEMENT, &c.
37
1. 4. from b. for
31.3.8.
32
nator, r. + P. ib. 1. ult. r.
1+m.
3º L. 3. P. 48. l. 3. for 1-x in the denomi-
32
1.
The method of transforming the expreffion in p. 56. may perhaps by fome be
thought tedious, as it is evidently reducible to 2 j X
=
which by divifon. in-
mediately becomes 2 j X -y-y³ − 1 + ; but this was done in order to fhew
the various tranfmutations an expreffion may paſs through before it arrives at the
moft fimple form, which is well known to be particularly uſeful in the finding of
fluents; for which reaſon I have alſo in ſome other places adopted this method.
P. 58. 1. 4. forr.; 1. 8. r. ß = +1. P. 60. 1. 5. draw a vin-
culum from the rad. fign in the num. P. 61. 1.
P. 61. 1. 7. for, — {y, r. — ¦ y². P. 63.
-ly².
l. 11, after numbers add―But, it muſt be obſerved, that this is underflood of the
expreffion in general, no particular relation being fuppofed to obtain between the co-
efficients of the fluentials. For, it is obvious, that if we can affuine, or diſcover,
fuch values of p, q, &c. as may by fubftitution make any of the coefficients with con-
trary figns become equal, if the difference of the exponents of the correſponding flu-
entials be a whole number when the figns of the propofed feries are +, or an even
number when they are alternately and, the algebraic fum will be had from the
+
fame formula, though fome of the values of *,, &c. be fractions in the former caſe,
or odd numbers in the latter. The feries in N° 265. Cafe 2. and 3. immediately
ariſe from the formula R in the above circumſtances; which are all the poffible cafes
in that theorem when x is 1.
P. 67. 1. 14, By an inadvertent omiffion (in the original) of the coefficient (§) of
the ſecond term, the latter part of this art. is rendered erroneous, let it therefore be
read as follows:- -We muſt therefore have recourſe to the œcumenical formula R
(art. 47.), in which fubftituting the above values of p, q, &c. there ariſes -
x3. x
· X
xx
플
​+4
+ 4 f = =
.x¹ x
>
expreffing the fum of the
1−x
given feries. Now put x = y, and the laſt expreffion is changed to
6
S
تودیو. و
6
3/
1 . y¹
1-y
—
+24, which by a proper reduction
نوو
becomes y—87'+1'+3 ƒ 232 + 24 ƒ 22 - 27 S 27 =
7
1
1 y
¹² y² −8 y³ + {j* + (4 L. 1+y+j³+y³ — 4L. 1+y+y² — 13 L. 1 + y²+y^ =
4L.
38
AS UP: P DE MEN T &c.
4 L. 1+ y³ +4 L.
Ity..
1+ y + y²
− L. y²+y*
- ¥ L• 1 + y² + y² = (becauſe
Ity
I+yɛ
is =
1+y+y²
2
1+ y²+
2
2
3
8 L. 1+ y³ - 27 L. 1+ y²+y², — 2√3. circ. arc, fin.
√3.Y
?
r. I.
2
(when y 1) 6+4L.4-27 L. 3-9 circ. arc, rad. 3, tang..
The fame is to be obſerved in the note p. *63.
2
:
P. 71. 1. 8. r. L. &c.; 1. 10. r. L. &c.; l. 13. r. —
✔2
8
2
4
342 L. &c. ; 1. 16.
За
dele L.a; l. 2. from b. for 2√2 L, &c. r. ✔2. L, and make the next term
3 a²
:
+; 1. ult. dele the firft term, and write t.
3
; fo will the fum of the feries be
4 +2
(P. 72. 1. 1. and 2.) 2 L.
√2+1
— §√2L · 4³ + 2³ + 1 +
I
L. 21 - 1
+
√2-1
3
1
一季 ​circ. arc, rad. 3
√
43
3
t.
(original).
4+43
P. *64. 1. 4. from b. r. L. 1+y'. P. *67. 1. 7. for £
£; l. ult. for Art.
x.
x, г.‹ƒxÞ : q†¤
r.
FK. P. 85. 1. 4.
L. 2, (origi-
55. r. in the preceding art. P. 77. l. 11. r. M; l. 12. for:x:+
P. 79. 1. 3. from b. for HEB, r. HBE; 1. 2. from b. for EK, r.
from b. forr. ; 1. ult. for 1, r. 1. P. 86. 1. 2. r. 48
巧
​nal). P. 96. 1. 17. add——in a pure algebraic expreſſion.
P. 99. 1. 3. from b. r.
45.2
-741
; 1. ult. r.
14
13.19
+
S
i+y³
3
14
3.13.19
j
X
I+Y
+
29
уў
2 j
I−y+ y² 1-y+y
and in the latter part of the example for 13 in the denom.
of the coefficient of the log. r. 39. and make the circ. arc affirm. and then 1. 6. p. 100.
will become 42
52
41
8
24T
117 arc, 45° rad. 1, +
14
39.19
L.2+ &c. (original).
P. 100. Ex. V. This example is rendered erroneous (in the original) by a miſtake
in fubftituting the values of p, q, &c. whereby the coefficient of the third term of
the fluential (1. 16.) is put down
I I
24
I I
Tб
تو
I I
inſtead of +36
I I
read !ì Sy² j− ¦¦ Sÿ + ¿ Sipas
S
; l. 21. r.
1+ y²
169
I I
36
let 1. 19. therefore be
+ & &c. and l. 24. for
+
A SUPPLEMENT,
39
1
&c.
}/ + { &c.
+ 12
27
P. 102. 1, 5. r. and
3 j
1+ y³
is evidently =
j
" ÿ
I−y+ y²
2j
+
1 −y+y² 3
1. 6. for latter r. fecond; l. 12. after the period,
add, To which adding (~-
(23)
I −y + y²
j
=)+
)+ circ. arc, rad. √3, t.
3 y
we have
4-2y
Si
35
− L.1+y~ { L. 1−y+y² + 2 circ. arc, rad.
√3
3 y
t.
And
1+y³
2
4 — 2
2 y
therefore &c.; I. 13. r.
14
3.13.19
P. 104. 1. 6. from b. r.
L. .2+ &c. P. 103. l. 1. for forms, r. terms.
2
3-y
1+y³
for . ×x, r. + ·×λ•
κλ. P. 110. Ex. V. for 16, 18, &c. in the denominators, r. 18, 22,
&c. P. 111. Ex. VI. By a mistake (in the original) in fubftituting the values of
P 9, &c. in the general formula, the coefficients in the expreffion for the ſum of this
and 1. 5. from b. for - }, r. + 2. P. 106. l. 12.
feries are rendered erroneous; let 1. 5. from b. therefore be written
135
3
S
I + √x
107 x² +
6
105
12
+.236
3
S
3
x x
; 1. 3. from b. for 110. г. 136, and
L. 2 -
3
2719
36
for 220, r. 272, then will the fum become
P. 136. In Order III. it may be neceffary to obferve that in (F) ≈ denotes the
number of terms, but in (S) z denotes the laft term, or m+zn. P. 137. 1. 7. write
the index at the end of the vincula. P. 140. 1. 7. for b d — ad, г. b d +ad. 1. 2.
r.
from b. for Zza +
x
2
Z
d,.r. Z X.za+
d, and for+, r. . P. 158.
2
2
1.2. from b. r. P. 159. 1. 4. and 5. from b. for even, uneven, r. equal, unequa!.
P..160. 1. 2. r. Order XXVI. Form 100. P. 163. 1. 5. from b. for 9-1 rq+1.
P.. 169. 1. 3, from b. r.
π-3
2. 16°
3
P. 170. 1. ult. for Ꭲ.
30 30
P. 184. 1. 6. for
ta, b, c (z) &c. r. abc (z). P. 188. 1. 9. for 5635, г. 5365; 1. 17. part of the ›
type having been diſplaced in working off this sheet feveral copies are faulty, it ſhould
be.
8.
8 . រៀន
+ + &c. P: 192. 1.7. after written add in. P. 200. 1. 2. in the
2
3.
numerators of the terms, г. 2 xº, 2². 1. 2 x¹¹, 2³ . 1 . 2 . 3 *
3
2³. 1. 2. 3 +¹³, &c.; 1. 7. r. 16.
+¹³,&c.;
P. 209.
40
&c.
A
SUPPLEMENT,
P. 209. I. 3. Mr. Simpfon is the author of the differential feries, from whence this
theorem is derived. P. 215. l. 4, r.
1.
1. ult. for x˜† ƒ,
풀
​r. x˜³ i ƒ.
X x
J
; 1. 6. after whatſoever, add if x be = 8;
As it has been objected that the fums of the feries Nos 229, 230, &c. are not eafily
attainable from the fluential expreffions there exhibited, I fhall therefore not only
give the fluents of thofe expreffions, but alfo fhew the investigation of them; as the
method is very extenfive, and applicable to the finding of a variety of other feries
with their fummations.
The fummation of the feries Nº 227. may be immediately deduced from the gene-
ral one No 108. or N° 111. where p, n, r are refpectively,, in the former,
or 1, 3, 1 in the latter; in which alſo v =´m = 2. Now multiply this feries and
t
its ſum by x, take the correct fluent, and multiply the whole by
arifes (by Form 35.)
x
and there
2
I
(N° 228.)
x +
4
1.5
4.6
x² + &c. =
-|
I
X
3x
S
*.
플
​2. 1−x
3 *
x-2
Multiply the fame feries (N° 227.) by xx, take the fluent, and multiply the whole
by
>
2
(N° 229.)
and we have (by Form 36.)
5+2
+
5.7+3
I
+
+ &c. =•
X :
3x²
5 4.7 4.6.9
3
—2A, s. √x, — † x².
·
The ſame ſeries being in like manner multiplied by xx, the fluent taken, and mul-
we have (by Form 38.)
tiplied by
5 x²
(N° 230.) — — *+
4.9
2
1
+ &c. = X
3 x x
I
3x²-xt
5
3A, 5.
x,-
? *¥.
And thus we may proceed as far as we pleafe, getting a new feries, and its fum, at
every operation.
Now, from the fums of thefe three feries is obtained the fum of N° 132. by means
of the general one N° 106. where b, c, d, &c. are refpe&tively 1,,
being the coefficients to the fucceffive powers of x in N° 227. Hence by ſubſtitution,
5.7
6"
5.7.9.&c.
"
4
4.6.8
&c.
A SUPPLEMENT,
&c.
41
I
&c. we have S&S=S
24x2
3
мен
1
x
I
2
*=
1 X
18x
I - X
I
3
2 x
X
2
3
x² - 2A +
I
x
3 x
2
5
X
12 XT
36 x 2
I-
rad. 1. fin. ✔x.
5
1
2
3A; the fluents
being ſuppoſed to be generated while x from o attains any given value not greater than
1, A being (as before) the circ. arc,
If we take x a root of the equation
punge the circular arc, and the fum of the feries
Now it appears that
N° 132. we have
I
+
I
ΤΖ
*¯* — ** o, we fhall evidently ex-
will then be expreffed algebraically.
is a root of this equation, confequently by fubftitution in
+
2
+
3
+&c.
5
5.7
5.7.9
2.4.5.7.2 2.4.6.7.9.2 2.4.6.8.9.11.2³ 2.4.6.8.10.11.13.21
= √. And by fubftituting other feries in N° 106. may the fummations of
a number of different feries be deduced, which will always become algebraical by
taking the ultimate value of x fuch as may exterminate the circular arc
G
REMARKS
42
A SUPPLEMENT, &c.
REMARKS on Mr. LANDEN's Obfervations on Converging Series.
W
HEN men of fuperior learning, and diftinguiſhed abilities, point out to the
world, with candour and good-nature, the real errors and miſtakes of others,
they certainly lay us under the higheſt obligations to them, and merit our warmneſt
acknowledgements; as an impartial enquiry after truth, wherever it is to be found,
is the great foundation of all ufeful knowledge. But when, under colour of fuch
animadverfions, we find the plain import of an author's words mifreprefented and per-
verted, and trifling, uneffential inaccuracies magnified into enormous errors, we have
the ſtrongeſt reaſons to fufpect that they have been dictated by a mind prepoſſeſſed
with prejudice, and actuated by difingenuoufnefs; and are therefore fo far from
having their defired effect, that they render the writers thereof ridiculous indeed to
every intelligent and unbiaffed reader. Whether Mr. Landen's Obfervations, with
reſpect to Mr. Lorgna's Treatife, ftand in the former or latter of theſe predicaments,
muſt be left to the determination of the candid and ingenuous, when they have atten-
tively confidered the following pages.
To be as conciſe as poffible in theſe Remarks, I ſhall confine myſelf chiefly to two
principal objections to what Mr. Landen has advanced in his Obfervations; which
are,
I.
1. That Mr. Landen's theorems cannot poffibly be deduced from what Mr. Simp-
fon has done in his Differtations without adopting the fame principles, or method of
procedure, as laid down in Mr. Lorgna's Treatiſe.
2. That the criterion exhibited by Mr. Lorgna of the poffibility of the algebraic
fummation of this clafs of feries, is not invalidated, or fhewn to be defective, by any
thing Mr. Landen has faid on the ſubject; but that he has, in his obfervations, ei-
ther mifunderſtood, or perverted, the fenfe thereof.
Now, the firſt objection is very evident from p. 4. (Obferv.) where the author tells
us, that before the following theorems can be deduced it is neceſſary to obferve that
I
x
1 I
is =
X
x
x
x
- I
**
(5-1) +
I
and therefore when sr is a pofitive integer the ferics will terminate, and the infinite
or
A SUPPLEMENT,
43
Sic.
or logarithmic part of the expreffion, to which the above equation refers, will be ex-
punged, when the fum of the coefficients is equal to o; or, as it is there expreffed,
when the indefinite quantity (*) is a root of the equation formed by making the fum
of thofe coefficients = o. But is not this the very principle upon which Mr. Lorgna
inveftig tes his formula for the algebraic fummation? For from this gentleman's
words we und.rftand, that when the fum of a feries is expreffed thus
Ax
S++ - BS + + (uſing Mr. Landen's literal fymbols) it will be att-in-
1 X
I - X
able in finite or algebraic terms whenever s-r is a (pofitive) whole number, and the
value of x ſuch that the coefficients of the fluents may be equal (which, in this caſe,
will always happen when x is = 1). And the reafon affigned is, becauſe, in theſe
x**
circumſtances, the above cxpreffion is equal A X
= A×
S¹
I X
I X
Х
vx, and
I X
I. X
1
X
=AX
x
***
is = 1 + x +² + +³ ( s − r) ; which are
the very requifites premifed by Mr. Landen for the algebraic fummation, this laſt
equation being evidently the fame with that given above, which is there defignedly
rendered obfcure by an unneceffary and awkward tranfpofition.
I fhall here juft obferve to the reader, that I am fenfible, it would ill become me,
were I ſo diſpoſed, to paſs any cenfure on the valuable works of fo eminent, worthy,
and respectable a character as was the late Mr. Thomas Simpfon; and would un-
doubtedly betray a moft ungenerous and unſcientific mind to offer to under-rate or mif-
repreſent them. But, on this occafion, I think I may with truth take the liberty to fay,
that it does not appear from the inveſtigations of Mr. Simplon's theorems (p. 62. Dif-
fert.), that he had any idea of the method given by Mr. Lorgna refpecting the algebraic
fummation of a feries*, as other principles were wanting for this purpofe which Mr.
S. had not premiſed; and therefore, that circumftance muft, by his procefs, be merely
adventitious. Neither does it appear that Mr. Simpfon even forefaw the ufe which
might be made of the propofition p. 146. Differt. in the fumming of feries; the
whole purport of that prop. being evidently only to reduce a compound fluxional ex-
preffion into fingle terms, in order to facilitate the finding of the fluent thereof.
And, in all probability, had not Mr. Lorgna publiſhed his treatife on Converging
Series, we ſhould never have feen Mr. Simplon's theorems put to the rack, in order
This, indeed, Mr. Emerfon has given a ſmall ſpecimen of in his Fluxions, p. 153. where he fhews onder
what conditions the expreffiona"
will become algebraical; and thereby fums the feries arifing from
expanding this expreffion, and taking the fluents of the terms. The formula Nº 2S6. is deduced from hence.
G 2
1 -X
to
44
&c.
A
SUPPLEMENT,
to extort a fecret from them, which, it plainly enough appears, they do not contain.
It muft, however, in juftice to Mr. Simpfon, be acknowledged, that the above-men-
tioned propofition is effent ally the fame (only not fo general in fome r.fpects) as
that which Mr. Lorgna has made the foundation of his method of fummation; but
the fuperftructure he has raiſed thereon has not the leaft fimilarity to any thing Mr.
S. has done. And this is all Mr. Landen can, with veracity, afcribe to Mr. S. in
the
7
* Whoever will take the trouble to compare Mr. Lorgna's theorems with Mr. Simpfon's prop. IV. and V. in
his Differtations (theſe being the only prop. in that treatiſe which relate to the fummation of this claſs of ſerics)
will find that they are intirely different from each other. Mr. S. in the former prop. exhibits the ſum of the
feries there propoſed by means of another affumed feries, whofe fum being given is denoted by S. This he ef-
fects by refolving the terms of the propofed feries into as many fimple fractions as there are factors in the deno-
minators of thoſe terms by the foregoing lemma, the firſt, ſecond, third, &c. terms of which refpectively con-
ſtitute other feries with a common coefficient to the terms, for each of which he obtains a finite expreffion by
fubtracting as many initial terms of the affumed or given feries from its fum S as are found to precede the initial
term of the ſeveral feries having a common coefficient, that is, till a term of S coincides with or becomes
-n
2
9
n
&c. which can happen only when the values of p, q, m, &c. are ſo related that m+zn may
>
9-n
m
be equal pn, q-, &c. or, &c. =z+1, which muft evidently be a whole number fince z is the index
of the number of terms lefs 1 of the feries S. And the proceſs in prop. V. is evidently by refolving the com-
pound coefficients of the terms of the propofed feries each into a number of fimple ones by the faid lemma, and
then ſuppoſing the fums of the ſeveral ſeries formed of the terms which have the fame coefficient to be each given.
But what fimilarity has all this to Mr. Lorgna's process? Had Mr. S. proceeded on Mr. Lorgna's principles,
when he had brought his feries (prop. IV.) to this form
A
X:
2
xp
xpt o
土
​z P
P
1 + n
+ &0.
B
z 9
zi + n
X:
土
​g+n
+ &c.
inflead of introducing the feries
expreffed it thus
z 9
&c.
m+r
+ &c. whofe fum muft neceffarily be given, he would have
978
א
土
​m+n
A
x P
S
11
I
F
B
29
S
x.9-
I
+ &c.
I F x"
which is a general expreffion for the fum of the feries, whatever the values of a, b, &c. p, q, &c. z, are.
And
the requifites for the algebraic fummation are (fuppofing, for example, only two terms to be taken, and x" put
be a whole (pofitive) number, an even number, or an odd number, ac-
A
B
x) that
may=
and
P
>
n
n
11
x
cording to the nature of the figns. That is, if the figns be all,
must be a whole number; if all +, an
11
odd
A
45
SUPPLEMENT,
&c.
the investigations of his theorems; the whole of what he has exhibited befide in his
Obfervations refpecting the fummation of this clafs of feries, being indifputably Mr.
Lorgna's.
It is then very evident from what has been ſaid above, that the theorems given in
Mr. Landen's Obfervations are inveftigated on the fame principles, and the fame con-
ditions we ſee are acknowledged to be abfolutely requifite for the algebraic fumma-
odd number; if - B and the other figns +, an even number. In other cafes, the fum of the feries will be infi-
nite, or expreffed only by circular arcs, or logarithms. For (in the cafe of two fluentials only) when the coef-
ficients are equal, and x" made = x, the above cxprcition becomes
A
P
n
nx
9-n
n
9-p
S
"
干
​干
​S
A
or
I F x
P
S
I
干​x
#
Xx
X
which will always be an algebraic quantity in the above-mentioned circumftances; the fluents being fuppofed to
be generated while x from o attains the value arifing from equating the coefficients of the fluentials. We do
not find any thing fimilar to this either expreffed or implied in the writings of Mr. Simpson.
Now the properties of theſe two Gentlemen's theorems are very obvious. Mr. Simpfoa's theorems (prop. IV.
and V.) cannot become algebraic expreffions unleſs S be exterminated, and this can only be effected in a genera!
way, when there are two more factors (at least) in the denominators of the terms of the feries than in the nu-
merators, and the ultimate value of x is 1, which appears from coroll. 3. p. 82. where the expreffion for the ſum
in that cafe becomes a pure algebraic quantity. From whence we thould naturally be led to conclude, that when
a propoſed ſeries does not come within thefe limitations, the fum is expreffible only by circular arcs, &c. And
in other cafes it is obfervable, that we are wholly uncertain refpecting the algebraic fummation, though the re-
lation of the factors come under the conditions fpecified; this being purely accidental, as has been before ob-
ferved. And when it happens that the fum of a feries is not attainable algebraically by these theorems, that is,
when the coefficient of S is fuch that we cannot difcover any value for x whereby it may be exterminated, there
is little or no advantage in introducing the feries S, unless we are furniſhed with a number of fluents that may
fuit any cafe propofed by the different values of m and n; without this it would be equally as expeditious to find
the fluents of the original fluentials themſelves as they arife in the theorem. And here it may be obſerved,
&c. happen to be fractional or negative numbers, that is, if p-n, q—n, &c. be o,
-m
-778
that when
n
1
or negative, S is no longer =
*
>
IF
but is of different values in the theorem, viz.
I
*
干
​&c. though this neceffary remark is omitted in Mr. Simpfon's note to prop. IV. And laftly, in
IF 17
theſe theorems, it is always neceflary that the factors in the terms of the propofed feries be in the fame arith.
progreffion, or be reduced to fuch before the fum can be exhibited thereby. But Mr. Lorgna's theorems are ge-
neral for this clafs of feries, no reſtrictions whatever being necetary for the gematric fummation,-
-the fums
of no given feries being previoutly requifite,—have the neceffary conditions pointed out for the algebraic fum-
mation; and that when the number of factors in the numerators are only one lefs than the number of thofe in
the denominators,————and lafly, are immediately adapted to feries whole factors are in the fame, or each in a
different arith. progreffion.
tion,
1
46
A SUPPLEMENT, &c.
tion, as thoſe laid down by Mr. Lorgna. And it will be equally as evident to any
one that will compare the formule in the beginning of this Supplement with thoſe in
the above-mentioned obfervations, that the latter are particular cafes of the former;
and therefore, in fuch cafes, the theorems are actually the fame. To put this matter,
however, beyond a doubt, let us take Mr. Lorgna's theorem, p. 17, &c.
177
111 p
n
9
x
I
λ
x
I
Xx
; which expreffion
nq·
m
n
by putting A =
Р
1
X
77
Р
I
1
X
n q
n
9
9
I
I
and B =
becomes
,
,
m
P
Р
n q
n q
n
n
9
9
P
111
卅一​7
Ax?
Bx
n
X
+
I
Xx
P
,
m
X9
n
X
tains the value denoted by X.
X"
the fluents being generated while x from o at-
Now this expreffion is evidently equal to
#
"
Р
9
р
m
9
9
n
X
X
Ax' .X
+ Bx X
X
I - X
that is, to
Р
9
m
m
172
n
n
n
Bx .X
- Bx X
X
I - X
P
m
m
m
m
"
n
X XAX +BX
9
X
S
9
9
n
*
x
n
X
-BX
* ;
1.
*
I
X
Р
9
m
P
X
is a pofitive whole number,
n
9
n
is =x
and confequently the above expreffion then becomes
x
9
x+
1+1 옴​+2
3
but when
+x (-4),
n
#
171
M
n
9
n
9
9
11
X XAX
+BX
X
BX
I - X
9X:
I
X
2.
X2
m
12
P + 9 + p + 29 + p + 39 ( 2 ).
p q
By
A SUPPLEMENT,
&c.
47
By the fame proceſs is the theorem p. 41. changed into
1*
p
9
X X:AX
X
-B
-C
X
m
m
+BX +CX X
I
I
4
X
X
P
q
X
I
p+q
+
7
+
X
+9x
I-
X
X² m
+
p+29
p+39 n
G 응​)
+ &c.
1
p+q
And in like manner may we proceed with the fubfequent theorems, till at laſt we
obtain by induction the general theorem p. 7. Supp.
Now, in the lemma (p. 16. Series) it is obvious that y may denote any algebraic
expreffion whatſoever in terms of x and known quantities; therefore, if we take
y =
x
P
9
1 - b x
-k"
Mr. Lorgna's theorem p. 17. becomes
m
n
9
1
x
X
m
I
m
I - b x
P
n q
S
•
n
12
4
9
I-ba
and the feries correſponding (expanding 1-6x by means of the binomial theorem) is
+I
11
x
p + q
m
燒
​b
-+3
+2
k. bc x
kbx
2
+
p+39 · m + 3n
&c.;
p + 29.m +211
and the laſt expreffion but one is evidently equal to
I
XX" X: AX
9
I
n9
I
I
A =
771
B =
P
772
P
77
11
9
x²+BX
S
な
​I
bx
, where
and X the ultimate value of x, as before.
=
But, it appears that the latter fluential
therefore when x (2
B
I
bx
*
is 1-x
g *, and
-) is a poſitive whole number the fluent (by Mr. Emer-
fin's
48
&c.
A SUPPLEMENT,
fon's 11th form) will be
(*),
P
9
9.
P
р
十分
​I
9
-1. ² + *−2 (*)
P
— + k + = + 1. — + k + = 2 + k − x (4) '6"
?
p
9
+π. A
+ I
xi-bx
9
+
p
+k+π+1.b
9
X
<fi
k
I-
-ba
x
b″
P
+7-1.B
9
+ 2 + x + x. 6x + 2 + k += -1.6 x
k bx
71
9
which (by writing any whole number for ) is evidently =
Р
1 +1. ² + 2(x)
9
+2.kt + 3 (*) b=
P
р
*+
9
9
Þ
k+
+2.6x
9
P
+ I
+1
P
+2
9
+
+
i.b
X:
S
1 - b x²x² x - x
9
9
I
XI-bx
k + ² + 2 + k + ² + 3 • b x²
9
9
P
+1
1+2
P
9
9
+2. +3
and confequently the laft expreffion (in this cafe) becomes
(π) ;
اه
9
m
Þ
བ།བླེ
P
P
+1.
+2 (*) BX
A
I
9
9
9
X
x : AX
+
X
S
9
k
I
bx
n q
Р
P
k +
9
9
+2.k+ +3•(π) bπ
р
P
+I
+ 2 (*) B. 1 — 6x+
k+
1+2.68
9
9
I
9
X
+
(*).
k +
서음
​+ 2 • * + 2 + 3 ( 7 ) b * X *
b*X
P
Р
P
+ I
+ I
+2
q
9
9
9
I
+ I
By the fame procedure is deduced the theorern in p. 8. Supp. Some other flux-
ionary expreffions may alſo be diſcovered whoſe fluents have the form requifite to the
conftruction of theorems fimilar to the above, by means of Mr. Emerfon's forms 11,
12, 13, 14; or by the propofitions in Mr. Simpſon's Differt. p. 149, &c.
Thus we ſee that thefe "narrow, embarraffed, ill-expreffed, particular formula" do,
by a moſt eaſy and obvious tranfmutation, exhibit theorems evidently more compre-
henſive than the all-comprizing Landenian-Simpſonian theorems, which have been po-
fitively afferted by Mr. Landen to contain at leaſt all the formulæ in Mr. Lorgna's
work (excepting thoſe in the IX Sect.), when, in fact, they do not comprehend one
fingle
A
49
SUPPLEMENT,
&.c.
fingle theorem in that treatife; even the general expreffions Mr. Landen has exhi-
bited in his obſervations for the fummations of thefe feries, being (as there expreſſed)
only particular cafes of the above formule, which, it is obvious, immediately re-
fult from thoſe given by Mr. Lorgna.
*
The fecond objection I have to make to Mr. Landen's Obfervations is, that the
criterion of the algebraic fummation of this claſs of ſeries exhibited by Mr. Lorgna,
is therein grofsly miſrepreſented; and therefore, that the Obfervator muſt either
have miſunderſtood, or defignedly perverted, the fenfe thereof.
Now the plain import of Mr. Lorgna's words are, 1. That if there be two fluen-
tial expreffions as A
S
equal to each other, and
x
P
9
I F
.
+ B
m
P
9
n
S
11
x
મ
X
IF X
whofe coefficients A and B are
(*) a whole (pofitive) number, the whole ex-
preffion will be reducible into finite or algebraic terms, fuppofing the higher figns to
take place. 2. If A be B, and the lower fign be used in the denominators only,
the whole expreffion will be finite when is an even number. 3. If A = B, and
the lower figns be uſed, the fame expreffion will be finite when is an odd number.
4. And if the higher fign be taken in the denominators, and the lower between the
fluentials, the whole expreffion will be infinite, whatever be the values of A, B,
and Now this is all very evident by reducing the above expreffion into
A
T
S
that
I- x
I-X
9
XX
al
KO
*, from whence we may eafily be convinced by divifion,
T
I x
will terminate in terms when is any whole number,
in terms
1 + x
* If, in Mr. Lorgna's theorems, we take b, e, &c. q, n, s, &c. each = 1, and a, e, &c. f, ™, r, &c. —
to A—1, C—1, &c. P—1, M—1, R—1, &c. reſpectively (ſuppoɓing P, M, R, &c. to be the factors in the
denominators of the terms of the feries, and A, C, &c. thofe in the numerators) we reduce them to the very
fame forms given by Mr. Landen, which are therefore evidently particular cafes only of Mr. Lorgna's theorems.
The peculiar advantage attending this laft gentleman's method is, that the theorems are immediately adapted to
feries whofe factors are in the fame or any orber arith. progreffion; whereas thoſe given by Mr. Landen are re-
ftricted to the fame progreffion, as well in the numerators as denominators, and we do not obtain the ſum of a
feries from thence till we have reduced each of the factors to the fame progreffion, by changing fome, or per-
haps all, of them into fractional numbers, which in various inftances will be attended with difficulty and awk.
ward fubftitutions.--Is this then the extenſiveneſs and ſuperior elegance which Mr. Landen would perfuade us is
to be found in bis method of expreffing theſe theorems? O Candour, Candour, Where doft thon hide
thy felf?
1
H
when
50
&c.
A SUPPLEMENT,
when it is an even number,
I + x
IX
7
I + x
in terms when it is an odd number, and that
1 + x
will not terminate, but run on fine fine in a feries, whatever the value of
may be. And as by this method of fummation the theoremns are always exhi-
bited by a combination or comparifon of each fluential with that formed from the
laft factor in the denominator of the feries, we may thence eafily deduce alb
the poffible ca es of the algebraic fummation, and the relation of the factors re-
quifite ther.to. For inftance, in N° 265 (which is the fame with that in order 14.)
if we equate the coefficient of the first fluential with that of the fifth, we fhall evi-
dently expunge the fecond and fixth terms, and confequently if w and ♪ be whole
numbers, the expreffion (2) will be finite whatever the values of and may be fepa-
d
rately, provided that x be taken of fuch value as to make the fum of the coefficients
become = 0. Again, if we equate the coefficients of the third and fifth fluentials,
and and w♪ be whole numbers, the expreffion () will be finite, Itt a and be
whatever they may feparately, provided x be taken as above *. And thus, by com-
bining or comparing the coefficients of the laſt fluentials with different figns, may all
the poffible cafes in any feries of this clafs be immediately diſcovered when the fum-
mation will be had algebraically, and literal theorems will arife by ſubſtitution in the
re'pective general formulæ in all ſuch caſes; which is done in the feries referred to
above (N° 265.). Now, if in this literal feries, we affume ph− e, q = 1, m = k,
n = 1, r = 2b—8—2e, s = 2, t = 2h+g, and u = 2; and ſubſtitute theſe values
in the expreffion (E) of the fi.ſt particular cafe, there will arife the ferics N° 272.
and the expreffion for its fum. In which feries if we take g = 1, and e≈ 1, we
And to theſe cafes it is a very eafy matter to adapt any number of feries at pleafure. For in the former
cafe, letn—de a whole number, then will a = e + 28, and therefore ♪ =
from whence
с
wte
π =
2
2
by affuming different values for c and e, and taking p, 9, &c. fuch as may correfpond to thoſe values, we get as
many feries as we pleaſe, that are exceptions to the general rule, and are yet fummible by the fame theorem.
As for inftance, let e I, and w 2, then is d , and 3.
and Ife 2, and w=3, then is
= ½.
d ¹, and π =
and ; let w 8, and e2, then is 3, 75. If w=.
=
7 =
If e = 1, and w = 4, then is d =
5
, &c. If the figns of the propoſed ſeries be +
=
T
6, and e4, then is 1, and 5, &c.
πλε
π
In the latter cafe; let w- de a whole number, then will w
>
and &= ; and by taking e and x
2
2
of certain values, we derive as before other feries adapted to theſe cafes.
is ♪ = 1, w = 3. &c. If = 4, e 2, then is = 1, w=3. &c. In all thefe inftances will the fums of
♪
the feries be attainable in finite terms, though ", ♪, &c. be fractions, the figns being +; or odd numbers, when
the figns are + and
Let(for instance) = 2, e1, then
obtain
A
St
SUPPLEMENT,
&c.
obtain Mr. Landen's theorem p. 28. Obferv. which therefore appears to be only a
particular cafe of that general theorem. Again, if we affume p = b−e+1, q = 1,
m = 2h+g+2, n = 2, r = 2b+g−2e+ 2, s = 2, t = b+g+1, u = 1; and
write thefe values in the fame feries, and alfo in the expreffion (E) of the ſecond par-
And in
ticular cafe, there will refult the feries N° 273. and its fum in finite terms.
this feries, if we take g = 2, e = 2, and multiply the whole by 4 (obferving the
remark when w- is negative) there arifes Mr. Landen's theorem p. 12. Obferv.
which alſo appears to be a particular cafe only of the feries N° 273. the expreffion
I
for the fum firft becoming by fubftitution X
48
-
1
I
}
+
b+ I 6+2
b+3
; and by reducing, and multiplying by 4,
26+4
I
I
I
12 X
X
I
26+2
b.b+1.b+2.6+3
Here then we fee plainly the grounds of Mr. Lorgna's algebraic fummation of this
clafs of feries. For the expreffion in p. 63. which Mr. Landen has laid fuch ftrefs
upon, evidently relates only to the general theorem when there is no given relation
ſuppoſed between the factors; nothing being more obvious than Mr. Lorgna's know-
ledge of the algebraic fummation depending on the different combinations of the
fluentials in the general formulæ, by his exhibiting the particular cafes when the
theorem in p. 96. is algebraically fummible; and his not particularizing thoſe caſes
in fome of the other theorems, can, at the worft, be called only an inadvertent
omiffion. Now, if Mr. Landen can make it appear, that the fluent of the binomial
expreffion before-mentioned can be had or exhibited in algebraic terms (that is, ex-
clufive of circular arcs or logarithms) in any other circumftances than thoſe we have
enumcrated, then will Mr. Lorgna's criterion fall to the ground; for upon no other
principle can it be invalidated or fhewn to be defective. But what fhall we fay when
we find Mr. Landen himſelf exprefsly acknowledging the impoffibility of it: For, in
his firſt corollary, he tells us, that the expreffion
I
(which is evidently
= A a³-
I
J-I
I
X
S
S
x
x
X
I FX
a
I
SBS) will be algebraical, when
s—r (that is 5—1—7—1) is a whole number, taking the higher figns; or an even
number, ufing the lower figns; but in other cafes the fluent will be aſſigned by circular
ares and logarithms. Here then has Mr. Landen come to the fame point with Mr.
Lorgna; having, we have before ſhewn, inveſtigated his theorems for the algebraic
fummation (or rather given us the fame theorems again, a little diſguiſed), and new
H 2
pointed
***
52
sxc.
A
SUPPLEMENT,
pointed out the poffibility of ſuch fummation, both on the very fame principles with
Mr. Lorgna! Now, would not any impartial perfon enquire with aſtoniſhment,
what could be Mr. Landen's motive in publifhing thoſe obfervations? Was it to fhew
the fallacy of the criterion above-mentioned, he has entirely defeated his own purpoſe by
miſtaking the principles; what he has faid thereon being, to the laſt degree, futile,
and frivolous indeed; and, in fhort, from what has been ſaid above, is evidently no-
thing at all to the purpoſe. Or was it, invidioufly to pluck the laurels from the head
of a man who appears to have fairly merited them, by publicly mifrepreſenting and
defaming his wok,this deſign is alſo rendered abortive: For the miſtakes
which Mr. Landen (and fome others) have found in Mr. Lorgna's treatiſe, are evi-
dent to every one not miſtakes in principle or effential errors, but only trifling inac-
curacies, or negligencies *, in the application of fome few examples to the general
theorems; which theorems are now univerfally acknowledged to be "exceedingly accu-
rate, extenfive, and ingenious." And with regard to the Commentator, I muft take the
liberty to inform the Obfervator, as it may be of fervice in future, that illiberal lan-
guage is not argument; and that the epithets oftentatious, boafting, vaunting, &c.
which he has fo profufely heaped upon him, appear to be ill-beftowed on the man
whoſe character is publicly known to be just the reverſe +.
Should Mr. Landen be diſpoſed to make any reply to what is advanced in theſe re-
marks, or to vindicate what he has already faid on the ſubject, it may not be impro-
per to inform him, that it will be expected he firſt of all fhew the poffibility of ob-
taining the fluent of the binomial expreffion ſo often mentioned, in finite or algebraic
terms, in any other cafe befide thoſe pointed out by Mr. Lorgna, and to confirm the
fame by examples. It is alſo wifhed he would give the reaſon why he has omitted
Mr. Lorgna's VII. Section; as alfo why he has not fummed one fingle ſeries alge-
braically by his theorems, when the number of factors in the numerators is only one
less than the number of thoſe in the denominators, the ſigns being alternately + and
and x = 1; or, as Mr. Landen expreffes it, when x is = I. But Mr. Lan-
den fays (p. 21. 22. Obferv.) that, in the above circumftance of the factors, it is
well known when the figns are +, the fum of fuch feries will be infinite; but if +
and alternately, the fum will be had in finite (algebraic) terms, x being ≈ 1.
Now this is exceedingly obvious from Mr. Lorgna's method of inveſtigation; the re-
2
y
*To this caufe we may certainly attribute the overfight in the expreffion L. ; for it cannot be fuppofed
that a perſon who could inveftigate the general theorems in that treatife, which are carried through ſuch a num-
I-Y
I
ber of intricate transformations, could be ignorant of ſo trifling a matter, as
being = to
I+y+ y²
† Vid. Hift. of Manchester, O&. Edit. Vol. IL
quifites
A
53
SUPPLEMENT,
&c
quifites in the latter cafe being cafily diſcovered by equating the coefficients of the
Auentials in the correfponding theorems. But as Mr. L. fays theſe things are well
known, it is to be wifhed that he would inform us, what Author befides Mr. Lorgna
hath ever ſhewn in a general manner the impoffibility of the fummation in the former
cafe, and the conditions or neceffary relation of the factors for the algebraic fumma-
tion in the latter.
And now, Gentle Reader, for your farther acquaintance with men, manners, and
things, I fball preſent you (by way of conclufion) with the tranflation of a few
pages from the Acta Eruditorum of Leipzig, for September, 1762 (p. 458.); and
when you have well digefted, and ftrictly compared it with that Gentleman's fecond
Memoir (p. 23, &c.), particularly the ninth and tenth articles thereof, I would
adviſe you to annihilate thoſe wonderful infignia with which that memoir is fo
profufely embellished; not only as a juft facrifice to the manes of John Francis de
Tufchis de Fagnano and his venerable father, but that the fcurrilous and il-natured
may not thereby take occafion to fay, that a British Mathematician had afcrited the
honour of a diſcovery to himſelf, which he had "bafely pilfered" from a Foreigner.
A Demonftration of a Theorem propofed in the Leipfig Acts, for January
1754 (p. 40.), by Archdeacon
rem,
JOHN FRANCIS DE TUSCHIS DE FAGNANO,
Ex Sancti Honorii Marchionibus.
(Tranflated from the Acta Eruditorum, 1762. p. 458.)
IN the above acts Geometricians are invited to demonftrate the following theo-
If in the ellipfis AEBF (Fig. 4.) whoſe principal axes are AB and EF,
any diameter be drawn as XY, and its femiconjugate CZ produced till CV is CA,
and from V a perpendicular be drawn to AC, the ellipfis will be fo cut in the point S
that the difference of the arcs XAS and YFS may be affigned geometrically. For
if from X the perpendicular XQ be drawn to CZ, we fhall have YFS-XAS =
2 CQ
In order to facilitate the demonftration of this theorem I fhall premiſe the fol-
lowing lemma;
In the above ellipfis let CA = a, EC = b, CM = x; then, if the abcifs CP (z.)
Σ
αν
be ſuppoſed to be ſo taken that its value may be expreſſed by
?
I fay
the
54
&c.
A SUPPLEMENT,
the arc EX-arc AT will be expreffed by
C
P - b xx
2
a³
a
2
2
a
My Father was the first who discovered this property, which, with its demon-
ftration, may be feen in the Italian Diary (Diarium Eruditorum Italia) Vol. 26. for
the year 1716, printed at Venice; being contained in the fixth article thereof, in-
tituled, A Theorem, from whence may be deduced a new measure of the arcs of the Ellipfe,
Hyperbola, and Cycloidt. It is also found in the fecond volume of his own works,
p. 336. printed at Peſaro (in Italy) 1750.
<
• Demonftration of the Theorem.
Having drawn VT, and (through M) XH, to AB, as alſo XQ 1 to CZ,
we have per conics (retaining the above notation) CO = √a²-x², CZ =
a
-b²
a² - b²
a²
a²
2
x², and CL=
; and becauſe of the parallels VP, ZO, ZC:
a
CO::VC: PC =
aV a² — x²
2
; alfo from the fimilar triangles CPV, CQL,
a
2
a
a
Q²
Ъг
Q
2
X
*
a
a - b²
a²
2
CV (CA): CP :: CL : CQ =
= (per lemma)
arc EX
-arc AT 1. Now the arcs XE, FY, HF are evidently equal to each
other, and therefore 2 arc EX arc HY; and the arc TAS is 2 arc AT; con-
feq. arc YH - arc TAS is = 2 CQ; to which adding arc HS
tranfpofing, becomes arc YH + arc HS
arc XSA = 2 CQ. Q. E. D.
arc TAS arc TX
arc XT, and
arc YFS -
Coroll. 1. To the point X draw the tangent XN, and
XN be parallel to the femidiameter CV, and KX evidently
YFS arc XAS is 2 KX.
=
‹ Cor. 2. When ≈ is =
CK
thereto, fo will
CQ; confeq. the arc
a
>
we have a² z
a² — b²
a²-b²
a²
x²z²=a*—a²x²,
a
This theorem, with its application to the rectifying of the Elliptic, Hyperbolic, and Cycloidal arcs (a copy
of which from the above Diary I have been lately favoured with) will be inferted in the fecond edition of my
Rationale of Circulating Numbers, which will foon be published, with additions.
*See Memoirs, art. 14.
Mr. Landen has alſo diſcovered that the arc EX
-arc AT is (CQ) the tang, XK, fee Mem. art. 9. 1
confeq.
1
i
A SUPPLEMENT, &c.
55
confeq. a² x x²+xa² is =
x 2 =
ax v a
2
a
2
X
,
a²
a² = b²
2
x²x²; and from hence arifes
a
z² + x²
√ a² = b²
a
as alſo from the above demon. arc YFS - arc XAS =
2. a²
62
X
2
+x²-a². Let now F be the focus of the ellipfe (Fig. 5.) then
a
will CF = √a²—62, and by ſubſtituting the linear values in this laft equation we have
arc YFS-arc XAS =
2CF
CA
× ✅CM² + CP² – CA²
=
2CF
× √ CM² — PV² =
CA
2 CF
X
√AOB—APB.
CA
Cor. 3. If in the analytical value of CQ we fubftitute the correfponding linear
CF2 CM.CO
values, there arifeth CQ =
X
CA²
CZ
Now from the foci F, f, draw
FX, ƒX, and produce the femiaxis CE till it meet the tangent NX in R; and let
XI be the radius of curvature of the ellipfe at the point X. Then (per conics) we
have theſe four equations
CZ = VFX.fX = VRX.XN = viX.XQ = VUA.CE.I
√FX.ƒX =
= IX.
Therefore alfo the following
CF2
MCO
X
CA
CZ
11
[CF2
X
CA
MCO
VFX.ƒX '
CF
MCO
X
CA
VRX.XN
CF2
MCO
CA² X
CF2
CA X
VIX.XQ
MCO
✔CA.CE. IX
which being multiplied by 2, are each equal to the arc YFS — arc XAS. And if in
this laft equation there be fubftituted
CA². LX3
CE4
for the radius of curvature IX, there
2 Cr
arifes are YFS - arc XAS =
X
CA 3
CE.CM.CO.
LX
• Cor. 4. If the difference of the elliptic arcs be required to be a maximum, we
have, by making the fluxion of (2CQ)
2xa² = b²
-b²
X
= o, and
Q
reducing
+
A SUPPLEMENT, &c.
56
3
a2
reducing, x =
Na+b
; which value of being fubftituted in the equation z =
3
2
ava' - x², we find z = x. If therefore we make x =
and take z = x,
a+b
the difference of the arcs YFS, XAS, will be a maximum.
Cor. 5. (Fig. 6.) When the point T falls in X, and theref. S in H, the equa
tion becomes, arc YH arc XAH =
2.a² = b²
as
X
a³
a+b
=2
2.a-b*.
• Cor. 6. (Fig. 4.) In the circumſtances mentioned in the two laft corollaries, the
tangent XK will be a maximum, per Cor. 1.
‹
Cor. 7. From Cor. 5. (or rather from the lemma) we have, arc EX - arc AX
= a-b; to which adding the quadrantal arc AE, there arifes, arc AE+ arc EX
arc AX = a- barc AE, that is, 2 arc EX — a—
= a−b + AE †.
- arc
Cor. 3. (Fig. 6.) Having drawn Xx, HY, parallel to AB, we have from the
preceding corollary, 2 arc HF = a−b + AF; and therefore 2 × arc EX + aic HY
— 2 . a−b + femiellipfe EAF, that is, arc XEx + arc HFY = AB-EF+EAF.
Which equation, multiplied by 2, exhibits the following theorem,
The whole periphery of the ellipfis EAFBE is equal to
2 X arc X Ex + arc HFY + EF¬AB.
Cor. 9. (Fig. 7.) In the cafe when the conjugate femidiameters CC, Cg, are
a
ava
2
2
equal, we have CM (x)
and therefore CP = z =
a²
62
2
a²
; hence (per lemma) arc EG - arc AT =
√ a² + b²
a²+b²
2
a² - b²
a² — b³
2CG
2.a+b²
11
arife the following analogies,
:: 2CG: a-b:: CG+Cg:
Cor. 10. From the equations in the Coroll.
5. and
9.
2
arc EX--arc AX : arc EG - arc AT ;; a−b:
2CG
AC + CE. ·
* And therefore arc FH. arc AH = a~b = KX (cor. 6.) fee Memoirs, art. 5.-
This very fame property alſo has Mr. Landen difcovered, fee Mem. art. 10,!!
F
INI
S.
E.
R R A T A.
In E N° 271. 1. 2. for gp, r. 89.
Σ
In E N° 275. for rad. 1, r. rad. r.
In F N° 288. forn-r, r. n
No
In E N° 293.
1. 3. for 2 C, r. c C.
In E Nº 300. for =b, r.
=b, г. = 2b.
In F. N317. for
In 2 No 295. write n over the exterior radical figns.
In F N° 311. for 5bc3, г. 46c³.
x2
x.
x3
r.
3
3
In F Nº 320. for
x5
xS
22)
t.
52
I.
U
Lately Published,
PON a new Plan, purely adapted to the Uſe of Schools, PRACTICAL
PERSPECTIVE; illuftrated with thirty-three Copper-Plates, and Moveable
Schemes, Vol. I. 8vo. (Price 5s. in Boards.)
2. THE RATIONALE OF CIRCULATING NUMBERS; with the Investigations of all
the Rules and peculiar Proceffes uſed in that Part of Decimal Arithmetic. To
which are added, Several curious Mathematical Queftions; with ufeful Remarks
on various Parts of the Mathematics. (Price 4s. in Boards. 8vo.)
3. A SUPPLEMENT TO PROFESSOR LORGNA'S SUMMATION OF SERIES. To which
are added, Remarks on Mr. Landen's Obfervations on the fame Subject. (Price
2s. 6d. 4to.)
4. A DISSERTATION ON THE SUMMATION OF INFINITE CONVERGING SERIES.
Tranflated from the Latin of A. M. Lorgna, Profeffor of Mathematics in the
Military College of Verona. With Notes, and an Appendix. (Price 10s. 6d.
in Boards, 4to. With the above Supplement, 135. in Boards.)
Alfs in the Prefs, and will ſhortly be publiſhed,
5. TABULE LINGUARUM.
Being a Set of Tables, exhibiting at one View the Declensions of Nouns, and Con-
jugations of l'erbs; with other Grammatical Requifites respectively effential to the Read-
ing of the following Languages, viz.
Latin
French
Italian
Spanish
Portugueſe
Teutonic or German
English
Sclavonian
Ruffian
Celtic or Erfe
Irish
Dutch
Polish
Scotch
Danish
Hungarian
Welsh
Swedish
Manks
Bohemian
The whole being intended to facilitate the Acquifition of any of thofe Languages, by
having at one View whatever is efteemed therein effentially neceffary to be committed
to Memory. The radical or ancient Languages being taken from the beſt Authori-
ties; and the derivative or modern, from the Determinations of the prefent Academies
and Literary Societies of the refpective Countries. And to each of theſe laſt is added
a fhort but comprehenfive Scheme of the prefent Pronunciation. To which is annex-
ed (at one View alfo) the moſt concife and approved Method of SHORT-HAND;
with Grammatical Abbreviations. Adapted to the English and French Languages.
HENRY
BY
CLARK E.
LONDON printed; and fold by J. MURRAY, N° 32. FLEET-STREET.
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