Summation of Series Dover Jolley

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SUMMATION OF SERIES

COLLECTED BY

L. B. W. JOLLEY, M.A. (CANTAB.), M.I.E.E.

SECOND REVISED EDITION

DOVER PUBLICATIONS, INC.

NEW YORK

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ternational Copyright Conventions.

Published in Canada by General Publishing Com- pany, Ltd., 30 Lesmill Road, Don Mills, Toronto, Ontario.

Published in the United Kingdom by Constable and Company, Ltd., 10 Orange Street, London WC 2.

This Dover edition, first published in 1961, is a revised and enlarged version of the work first published by Chapman & Hall, Ltd., in 1925.

Libray of Congress Catalog Card Number: 61-65274 International Standard Book Number: O~#W-(ioo23-8

Manufactured in the United States of America Dover Publications, Inc.

190 Varick Street New York. N. Y. 19914

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PREFACE TO DOVER EDITION

A SECOND edition published in the United States of America provides an opportunity for including many new series with an increase of more than 50 per cent over the original number. It has also been possible to rearrange the series in a more reasonable form.

Some corrections have been received from readers, and useful suggestions have been made by them for the present arrangement.

These are gratefully acknowledged, and it will be of great assist- ance for future editions if readers will communicate their ideas for further expansion.

In using this collection himself, the author has experienced difficulty in tracing certain series, and there does not seem any solution excepting a complete search through all the series given.

For example, certain series including inverse products appear in different parts of the book, and, if a search is to be avoided, a complete rearrangement combined with excessive duplication would be necessary. It does not seem possible, as in the case of a collection of integrals, to arrange them in a completely rational manner. Any suggestion in this direction wouId be specially welcome.

Among the new series included are some of those developed by Glaisher in many publications, notably the Quarterly Journal of Mathematics. Basing his series on Bernoulli functions, Glaisher evolved a number of coefficients which apparently simplify the appearance of the series. In this present collection, only a few are given, and the original articles should be consulted if the reader wishes to investigate them further.

The author wishes to acknowledge permission by the London Scientific Computing Service to publish tables from the Index of Mathematical Tables by Fletcher, Miller, and Rosenhead-an

V

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exceedingly useful book for any engaged in work on applied mathematics; and also to thank Mrs. H. M. Cooper for her excellent work in typing a difficult manuscript.

L. B. W. JOLLEY

623 Upper Richmond Road West, Richmond, Surrey, 1960

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PREFACE TO FIRST EDITION

FOR a long time past there has been a need for a collection of series into one small volume for easy reference together with a bibliography indicating at least one of the textbooks to which reference could be made in case of doubt as to accuracy or to the method by which the series was arrived at.

The 700-odd series in this collection (with the exception of a few which have been specially prepared) are not new, and repre- sent only the labour of extracting the material from the many textbooks on algebra, trigonometry, calculus and the like. Yet such a collection will, it is felt, be of considerable benefit to those engaged in the solution of technical problems, and will save a great deal of time in searching for the required result.

Criticism may be offered on the grounds that the inclusion of easy algebraical summations is unnecessary, but they have been inserted for a very definite purpose. For example, a series of inverse products may have for its sum an expression which is simple to find; but on the other hand, the solution may entail a complicated expression involving the integration or differentiation of other series. For this reason the arrangement of the series has been difficult, and overlapping is unavoidable in certain instances.

To overcome this dficulty, the series have been set forth in as pictorial a manner as possible, so that the form of the individual terms can be readily seen.

On this account also, the inclusion of such series as are evolved for elliptic integrals, Bessel functions and the like has been restricted, perhaps to too great an extent; but reference to standard works is usually essential in such cases, and practically only such references are included.

The final column refers to the bibliography at the beginning of the book, and here again it has been quite impossible for obvious reasons to provide for all the references.

vii

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One of the most useful works, if it is desired to pursue any one particular problem further, is the Smithsonian Tables.

The scope of many of the series can be greatly enlarged by differentiation or integration of some of the forms given, and in the case of an integrated series, the constant of integration must be obtained by suitable methods. Infinite products are often of value in obtaining new series by taking logarithms and by differentiating or integrating subsequently.

In many cases it has been impossible in this small volume to comment on the limits or assumptions made in any particular summation; particularly is this the case with oscillating series:

and in case of doubt it is always safer to refer to a textbook, and to bear in mind that this collection is supplementary to, and not in place of, the usual mathematical books.

Special attention is drawn in cases of difficult summations to the General and Special Forms (pages 216-225).

In all cases logh denotes the logarithm to the Napierian base, in accordance with modern practice.

Finally, any additions or corrections would be welcomed for embodiment in subsequent editions.

L. B. W. JOLLEY

Fairdene, Sheen Road, Richmond, Surrey, 1925

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BIBLIOGRAPHY

Indicating Author Title and Publisher

Letter At

B C

D E F G H J K L M N

T. J. Bromwich L. L. Smail G. Chrystal

Levett and Davi- son

S. L. Loney H. S. Hall and S. R. Knight E. T. Whittaker and G. Robinson H. Lamb L. Todhunter C. P. Steinmetz J. Edwards J. Edwards G. S. Carr

Introduction to the Theory of Infinite Series, London : Macmillan Co., 1926.

Elements of the Theory of Infinite Processes, New York: McGraw- Hill Book Co., 1923.

Algebra, An Elementary Text Book for the Higher Classes of Second-

ary Schools, New York: Dover Publications, Inc., 1961.

Plane Trigonometry, New York:

Macmillan Co., 1892.

Plane Trigonometry (Parts I and II), Cambridge : Cambridge Univer- sity Press, 1900.

Higher Algebra, London: Mac- millan Co., 1899.

Calculus of Observations, Glasgow : Blackie and Son, 1937.

Infinitesimal Calculi, Cambridge : Cambridge University Press, 1921.

Integral Calculus, London : Mac- millan Co., 1880.

Engineering Mathematics, New York: McGraw-Hill Book Co., 1911.

Dtflerential Calculus for Beginners, London: Macmillan Co., 1899.

Integral Calculus for Beginners, London: Macmillan Co., 1898.

Synopsis of Pure Mathematics, London : Hodgson, 1886.

t In the text, the numbers preceding the Reference letter refer to the volume of the work cited; the numbers following the Reference letter refer to pages.

xi

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Author

0 E. W. Hobson P

Q R

E. T. Whittaker and G. N. Watson E. Goursat T E. P. Adams U W. E. Byerly W G. Boole X A. Eagle Y J. Edwards Z J. Edwards AA K. Knopp AB H. S. Carslaw AC Fletcher, Miller,

and Rosenhead AD

AE AE.1.

AG

E. Jahnke and F.

Emde

J. W. L. Glaisher J. W. L. Glaisher E. W. Hobson

Title and Publisher

A Treatise on Plane Trigonometry, New York: Dover Publications, Inc., 1957.

Encyclopaedia Britannica, 11 th edition.

Modern Analysis, Cambridge: Cam- bridge University Press, 1920.

A Course in Mathematical Analysis, Vol. 1, New York: Dover Pub- lications, Inc., 1959.

Smithsonian Mathematical Formu- lae, Washington : Smithsonian Institute, 1922.

Fourier’s Series, New York: Dover Publications, Inc., 1959.

Calculus of Finite Differences, NewYork: Dover, 1960.

Fourier’s Theorem, New York:

Longmans Green and Co., 1925.

Differential Calculus, London: Mac- millan Co., 1938.

Integral Calculus, Vols. I and II, London: Macmillan Co., 1922.

Theory and Applications of .Infinite ,SS~~g2&sgow : Blackre and Fouritk’s Series and Integrals, New Tgyr : Dover Publications, Inc.,

.

Index of Mathematical Tables, Lon- don: Scientific Computing Ser- vice, 1946.

Tables of Functions, New York:

Dover Publications, Inc., 1945.

Quarterly Journal of Mathematics, Vol. 29, 1898.

Quarterly Journal of Mathematics, Vol. 28, 1896.

The Theory of Functions of a Real Variable, Vol. II, New York:

Dover Publications, Inc., 1957.

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SUMMATION OF SERIES

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Series No.

I. Arithmetical Progression

(1) a + (a + d) + (a + 24 + . . . n terms II. Geometrical Progression

(2) a + ur + ur2 + . . . n terms (3) a + ar + ar2 + . . . co

(4) 1 + ax + a2x2 + ~3x3 + . . . co

III. Arithmetical and Geometrical Progression (5) a + (a + d)r + (a + 2d)r2 + . . . n terms (6) a + (a + d)r + (a + 2d)r2 + . . . co (7) 1 + 2x + 3x2 f 4x3 + . . . co

(8) 1 + 5 + 52 + g+... 4 7 ⁵³ n terms

2 3

(9) l + 2 + F + 23 ff_+... n terms

3 5 7

(10) 1 -I- 2 + 3 + g + . . . co (11) 1+3x+6x2+10x3+...co t See footnote to Bibliography.

(13)

ARITHMETICAL, GEOMETRICAL PROGRESSION 3

Referencet

= g (2a + (n - l)d} = ; (a + I) where I = last term F. 29

= p - 1) r-1

= - a where r < 1 l-r

= - 1 where ax < 1 I - ax

a

=rr+ dr(1 - P-l) _ {a + (n - l)d)P (1 - r)* 1 -r

1

= (1 - x)2 where x < 1

=g 12n+7

16 --igqi=i

=4-~* 1 -- 2nEl

=6

= - 1 where x < 1 (1 - x)’

F. 39 F. 40 F. 158

F. 44 F. 44 F. 44 F. 45

F. 45

F. 45 F. 45

(14)

Series No.

(12) x(x + y) + x*(x* + y*) + x3(x3 + ~3) + . . . n terms

2 3 2 3 2

(13) 3 + p + jj + jyi + jj +. . . co

(14) 7” - ; +; - ;4 +... co

32 52 72

(15) 12+Z+22+Zj+...nterms

(16) 1 + 3x + 5x2 + . . . + (2n - 1)x”-’

IV. Powers of Natural Numbers

(18) 1 + 2 + 3 f 4 +. . . n

(19) 12 + 22 -t 32 + 42 + . . . n*

(20) 13 + 23 + 33 + 43 + . . . n3

(21) 14 + 24 + 34 + 44 + . . . n4

(15)

POWERS OF NATURAL NUMBERS xyx2n - 1)

= x2 - 1

+ X.Y(X”y” - 1) xy - 1

=- 9 8

=- 23 48

5

Reference

F. 46

F. 46 F. 46

= 34 - (4n2 + 12n + 17)&

= 1 + x - (2n + 1)X” + (2n - 1)x”+’

(1 - x)2

nP+l

=- p+l 1 P

+is 5 0 B3nn-5 -. . . where ’

0 are the binomial coefficients n

and B, are Bernoulli numbers, see No. (1129). The series ends with the term in n if p is even, and with the term in n2 if

p is odd. T. 27

= n(n + 1) 2

= n(n + 1)(2n + 1) 6

= (““: Y2

= & n(n + 1)(2n + 1)(3n2 + 3n ^-

F. 50 F. 50 F. 51

1) F. 256

(16)

Series No.

(22) 15 + 25 + 3’ + 4’ + . . . .s (23) 16 + 26 + 36 + 46 + . . . n6 (24) 1’ + 27 + 37 + 47 + . . . n’

(25) 12 + 32 + 52 + 72 + . . . n terms (26) 13 + 33 + 53 + 73 + . . . n terms (27) 13 i- (1.5)3 + 23 + (2.5)3 + . . . (28) 22 + 42 + 62 + 82 -t- . . . n terms (29) 12.21 + 22.22 + 32.23 +. . . n.terms (30) 1.22 + 2-32 + 3.42 -t . . . n terms

(31) (n2 - 12) + 2(n2 - 22) + 3(n2 - 32) + . . . n terms (32) 2 (2n - 1)2

1

(33) 2 (2n - I)3 I

(17)

POWERS OF NATURAL NUMBERS

f n(4n2 - 1) n2(2n2 - 1)

1 (n + l)(n + 2) 2 1

s 1 >

--

2 8

2n(n + 1)(2n + 1) 3

2”{2n2 - 4n + 6) - 6 j!j n(n + l)(n + 2)(3n + 5) + n2(n2 - 1)

f n(2n - 1)(2n + 1) n2(2n2 - 1)

xyx2n - 1)

= x2 - 1 _- i n(n + 1)(2n + 1)

= x2(x& - 1) _ Y2(Y2” - 1) x2 - 1 y2 - 1

7 Reference

F. 337 F. 338 F. 338 F. 256 F. 256

F. 256 F. 323

= GYP - 1Mv)“+ + 1) _ {(;)” - I){Q+l + 1);

(XY)“(XY - 1)

u(l - 9

(18)

Series No.

(38) 2 (a,,~ + qw-1 + . . . a,) I

See No. (17)

V. Products of Natural Numbers

(42) To find the sum of n terms of a series, each term of which is composed of r factors in arithmetical progression, the first factors of the several terms being in the same arithmetical progression: Write down the nth term, affix the next factor at the end, divide by the number of factors thus increased, and by the common difference, and add a constant.

(43) 1.3.5 + 3.5.7 + 5.7.9 +. . . n terms

(44) 1.2+2.3+3.4...nterms

(45) 2.5 + 5.8 + 8.11 +... nterms

(46) 2.2 + 4.4 + 7.8 + 11.16 + 16.32 . . . n terms (47) 1.2.3 + 2.3.4 + 3.4.5 . . . n terms

(48) 1.2.3.4 + 2.3.4.5 +. . . n terms

(19)

PRODUCTS OF NATURAL x2(1 - X”) Y2(1 - Y9

= (x - Y)U - xl - (x - Y)(l - Y)

= {ul$, + qb,-1 + . . . + upz}

b,= lr+21+3r+...nr

=ex where x c 1

NUMBERS 9

Reference

= (1 _” x)2 where x < 1 1 - X” n(n + 3)x

= (1 - 2(1 - x)2

+ n(n + 1)x”+’

2(1 - x)2

F. 314

= @J - l)@ + 1m + 3)(2n + 5) + y 4.2

= n(2n3 + 8122 + 7n - 2)

= n(n + l)(n + 2) 3

= n(3n2 + 6n + 1)

= (n2 - n + 4)2” - 4

= f n(n + l)(n + 2)(n + 3)

= 5 n(n + l)(n + 2)(n + 3)(n + 4)

F. 315 F. 52 F. 318 F. 333 F. 322 F. 322

(20)

Series No.

(49) 1.4.7 + 4.7.10 + 7.10.13 +. . . n terms

(50) 1.4.7 + 2.5.8 + 3.6.9 +. . . n terms

(51) 1.5.9 + 2.6~10 + 3.7.11 +. . . n terms (52) 6.9 + 12.21 + 20.37 + 30.57 + . . . n terms

nth term is (n + l)(n + 2)(2n* + 6n + 1)

(53) 2.2 + 6.4 + 12.8 + 20.16 + 30.32 + . . . n terms nth term is n(n + 1)2n

(54) 1.3.22 + 2.4.32 + 3.5.42 + . . . n terms (55)

(56)

i: (P - 4(q - 4 1

n m(m+l)...(m+n- 1)

c tZ!

I

(57) n b(b + I)@ + 2)...(6 + n - 1)

c

l a(a + I)(u + 2). . .(a + n - 1) (58) 12.X 2*.x*

m + n . . . n terms This series is integrable if x = 4.

(59) n c

n24n l (n + l)(n + 2)

(21)

PRODUCTS OF NATURAL NUMBERS I1

Reference

= ; (3n - 2)(3n + 1)(3n + 4)(3n + 7) + ; F. 322

= &(n + l)(n + 6)(n + 7) F. 322

= $n(n + l)(n + 8)(n + 9) F. 322

= g ncn + l>(n + 2)(n + 3)(n + 4) + ; (n + l)(n + 2)(n + 3) - 2

= ($ - n + 2)2n+’ - 4

= h n(n + l)(n + 2)(n + 3)(2n + 3)

F. 331 F. 332

F. 323

=“(n6+l){(2n+ l)-3(p+q)}+npq

= (m + l)(m + 2). . . (m + 1 + n - 1) _ 1

n! c. 200

=(m+nY-l

m! n!

b(b + l)(b + 2). . .(b + n) b

= (b + 1 - u)a(u + l)(a + 2). . .(a + n - 1) - b + 1 - a T. 28

qn+’ n - 1 2

= -.-

3 n+2+5

W. 58

w. 58

(22)

Series No.

VI. Figurate and Polygonal Numbers (60) F&rate numbers-

1 1 1 1 1 1 .,.

123 4 5 6 .*.

1 3 6 10 15 21 . . .

1 4 10 20 35 56.. .

1 5 15 35 70 126 . . .

The sum to n terms of the rth order (61) Method of Differences+ d LX & K@w-~& k~.- ~?rclikZ

One Series is 12 40 90 168 280 432 . . .

1st Diff. 28 50 78 112 152 . . .

2nd Diff. 22 28 34 40 . . .

3rd Diff. 6 6 6 . . .

4th Diff. 0 0

The nth term is The sum

(62) 4 + 14 + 30 + 52 + 80 + 114 + . . . n terms (63) 8 + 26 + 54 + 92 + 140 + 198 + . . . II terms

(64) 9 + 16 + 29 + 54 + . . . n terms (65) 4 + 13 + 35 + 94 + 262 + . . . n terms

(66) 2 + 12 + 36 + 80 + 150 + 252 + . . . II terms

(23)

FIGURATE AND POLYGONAL NUMBERS 13

Reference

= 6(2n - 1) + ; n(n + 5) F. 333

= ; (3n - 1) + ; n(n + l)(n + 5) - n F. 333

= ; n(n + l)(n + 2)(3n + 1) F. 332

(24)

Series No.

(67) 30 + 144 + 420 + 960 + 1890 + . . . n terms (68) 2 + 5 + 13 + 35 + . . . n terms

(69) 2 + 7x + 25x2 + 91x3 + . . . n terms

VII. Inverse Natural Numbers

” 1 (70)&=C+loghn+&&)

1

- n(n+ $(n + 2) -***

Also 1 + ; + . . . + t

1 1 1 1

(71) 1 - 2 + 3 - 3 + 5 - . . . co

1 1 1 1

(72) 1 - z - 5 + s + 3 - . . . a~

(73) (1 - ; - ;) + (; - ‘6 - ;) + (4 - & - A) +. . .a3

(74) -- 5 1.2.3 + 4.5.6 14 +*-* O”

(75) 2( 1 - ; + f - $ + ; - . . . 00

1 1 1

= --- 1.2.3 + 5.6.7 + 9.10.11 +“-O”

(25)

INVERSE NATURAL NUMBERS

= & n(n + l>(n + 2)(n + 3)(4n + 21)

15 Reference

F. 332

= ; (311 - 1) + 2n - I 1 - 4X” 1 - 3”X”

= 1-4x + I-3x

F. 272 F. 272

where C = Euler’s constant, see No. (1132),

1 1 19 9

u2 = E a3 = fi a4 = 120 a5 = To

x(1 - x)(2 - x). . .(k - 1 T. 27

= 7.48547 where n = 1000

= 14.39273 where n = 106

= logh 2 zz l-06 ‘L -L IA 52

= 0.43882 = E - - 1 logh.2

4 2

= ; logh 2

m

c

^{9n - 4}

= , (3n - 2)(3n - 1)(3n) = logh 3

A. 325

F. 195 H. 475

C. 252

C. 253

C. 252

(26)

Series No.

(76) 1 + 1 - 1 - 1 + 1 + 1 - . . . a 5 3 5 p n

(77) 1 - 1 + 1 - 1 + 1 - . . . co j 3 5 g

1 1 1

= 1 -2 [ c5+r9+ll.13+“‘a 3

(78) 1 -;+;-&+A-... 00 (79) 1 - ; + ; - & + . . . a3

(80) 1 z - 5 + 1 - 1 8 i-i 1 + . . . co

(81) 1 - 1 2 + 1 4-;++-;+... a

(82) 1 - ; + ; - h + . . . co

(83) I-;-;+;+A-;-~+...ocI

(84) 1 - ; + f - h + h - h + . . . 00

(85) 1 -;++$f-... co

1 1.2 1.2.3

(86) 1 - 3 + n - 3*5*7 +... aJ

(27)

INVERSE NATURAL NUMBERS 17

Reference

5\\07c!Yo735 c. 335

= f (1 + 2/2) 0,4cl8oLsYqc1y~ M.132

=3 ( 22 1 43 + logh2

0,%3~.6c1

E3w3 A. 189

6,373

55022..77 A. 189

=&

0, &oq sqqyag(-J 27 p- 281

= +2 [m +2 logh (d/2 + 1)] o,%ddq=q=‘Y A. 190

= 5 logh (2 + 1/3) 6,7(;>0%5% t AB. 166

\

= logh (1 + 1/2)

= $logh(y)

A. 528 Y. 90 Y. 90

= ; (C + logh n) + logh 2 + sz - (23;n!)Bz + . . .

For C see No. (I 132) and for B,, see No. (1129) A. 325

(28)

Series No.

2x2

1+x+1+x2+

(88) A- - - 1 + 4x4 X4 -*- a3 (89) s?- 1 _ X2 + 1 _ A X4 + 1 _ Xl? + * * * n mms m?!-

(90) x L - x4

l-x2+l-x4+l--x8 +“*O”

(91) J-- + -& + -& . . . co x+1

(92) 1 ‘;yx2 + 1”“,“;‘:4 + 14~3x;~;8 +... GQ (93) ^l ^[Jj ^-

(94) f $2 - ; $4 + f l&j - . . . co

(95) ;.(k2) +&j(j+)l+&(&)5+... * (96) 2 2x {-$ E-’ - & SE}

1

A=-;; B=-

(n T21)2 ^c= ^-9 VIII. Exponential and Logarithmic Se&s (97) 1 + ax + fg + fg + . . . co

.

(98) 1 + x logh a + (x 1’;: u)2 + . . . co

(99) x - 5 + $ - . . . co =x(1 - x) + +x2(1 - x2) + . ..a3

(29)

EXPONENTIAL AND LOGARITHMIC SERIES 19 Reference

= yx X where (x2 c 1) T. 118

1

1 1

=--zn l-x A. 24

= ex where x2 < 1, and

= -- 1

x-l where x > 1 T. 118

= -& where x2 > 1 T. 118

1 + 2x

=1+x+x2 where x < 1 Y. 54

1 1

x(x + 1) + x(x + 1)(x + 2) + * . . c0 A. 102 7rlogh2

=- 8 where s2,, 1 1 1

= 1+2+3+...+5

A. 192

= x where 1x1 < 1 Q. 132

AB. 167

=P

F. 188

=logh(l +x) wherex< 1 F. 191

F. 188

(30)

Series No.

(loo) -x-T-g-... a3

(101) 1 + 23 + ; + ; + $ + . . . al

. . .

3

(102) ‘2 - A2 + & - A4 + . . . m (103) 2

1 ;+;+;+...m

. . . >

(lo4) 5n(4nz1- 1)2 1

(105) (x - 1) - f (x - 1)2 + f (x - 1)3 - . . . co

2 3

(106) i + g + . ;+;+y . 18 +... a3 .

(107) c (1 + x + 4 x2 + 4 x3 + g x4 + . . . m)

(108) 1 +2~+7,2.~.;+... co

(110) 2641 +;)*3+;(1 +;++4- . . . co}

(111) 2~+;(;+++:(1 +;++‘+ . . . co}

(112)d+(l-;+f)~l-f+t-;+;)f+ . ..a

_- y

(lls)*~+(l-f+~)~+(l-~+~-;+~)~+ . ..a

(31)

EXPONENTIAL AND LOGARITHMIC SERIES 21

Reference

= logh (1 - x) where x < 1 F. 191

= 15c F. 339

= logh 3 - logh 2 F. 195

=C -1

F. 196

=-- 3

2 2 logh 2 C. 253

= logh x where 0 < x < 2

= 3(c - 1) =7

= &

= (1 + X)@ F. 338

= ,x - logh (1 + x) F. 338

= {logh (1 + x)}* where x < 1 F. 191

= [logh (1 - x)12 A. 191

= - logh(l + x).logh(l - x) A. 191

= $ (tan-l x) logh E A. 191

(32)

Series No.

- (114) e-m + e-gn + &SW + . . . Co

(116) 1 -;(l+;)+f(l+f+;)-...m 017)

s-&

1

(118) zz

(119)

25

1 -

(120) Zn(n-, 1)

J

(12l)x+(l+;)x2+(l+;+f)x3+...m w (n

(122) l&n + ;;;;

(123) $‘l’ + 23 +,‘!’ +. . . n3) x”

I

$ f/7m &&

(125) ~(-w&

I

(33)

= v4 - W(b) 21 l/4,3/4

r (4) = 3.6256 T. 144

= l A where A = c* and a,,+l

= ( a, + an-l + . . .

=- 7; - ; (logh 2>* 0, S022LtO5UJ

Y. 111 A. 520

= logh s H. 460

= logh+* where x < 1

= kx logh Gx where x < 1

= ((x2 - 3x + 3)e - 3x2 - 3)

X2

(

⁷

= l x + j x* + 2x3 + ; x4 1 77 l n-= + z-- 1

= -. --

2x @Lx - E-nx 2x*

77 1 1

= -. --

x @Ix - E-“x zx2

F. 338

C. 236

C. 235

T. 135

(34)

Series No.

(126) & + & + k2 + ...A +&

-~(+2-$)+&j($&i)-j&(&$) +&$&-J-J+...20

(127) f logh n + logh (n + 1) + . . . +logh (m - 1) + $ logh m - &(:-g+&($-;)

-&&-$--g+...m (128) 2 [n logh (2#) - l]

I

(129)x++~x~+... co

(130) 1 + (x0) + $ (XC-y + $(XP)’ + $(xc-)4 + . . . co

= 1 + 2 cn “,t’“-’ (xe-x)”

x3 x5 x7

I

(131) m + c5 + c7 +. . . 00 (if convergent) (132)t 1 - 2(2 - lp, ; + 2(23 - 1)~~ $

- 2(25 - l)B,g X6 +. . . co

(133)t1+2ux+(2C7)+2(~+x)+;B,}+yB2(;+x)

+qgE4(k+x) 4B2}+... 00

t For values of B,,(x), see No. (1146).

(35)

Reference

= logh t where m and n integers x. 141

= m logh !f - n logh : where m and n integers

= ; (1 - logh 2)

logh {x + 2/l + x2} where 1x1 Q 1

x. 141 A. 526 A. 197

= ,p Y. 456

1Z. 165

= 2x

l -x - c-x AE. 12

24&x

=- e - c-0 AE. 14

(36)

Series No.

(134H ;+a(;) +$?&) +&(;) +... ol)

(135)t f + 3xB2 (;) + y BJ (f) + y B4 (;) + . . . 00

(136)+ ;+6xB+)+yB3(;)+... co

(137) x - (1 + $9 + (1 +; + +3

- (

1 1 1

1+j+j+a ++...co

) (138) x-2’y+2.4 ^{1 x3} -.- I.3 x5 5 - . . . co

(139) logh2 + &;x2- &&x4+ Xx6-... 00

. . . . .

(140) 1 -+---**. x2 x3

12 24 a =

s : (1 - x)’ dt x3 2 x5 2-4 x7 -.--

(l41)x+5-3*T+3*5 7 +** 00

2 2.4

(142)x--x3+-x5+... co

3 3.5

(143) x’ - ?.x’ + 21_4 x” + 2 3 4 3.5’6 *-* cQ

W) x“ x5 x6 x’ $3

(145) $ + y + y + . . . co t For values of B&c), see No. (1146).

(37)

EXPONENTIAL AND LOGARITHMIC SERIES 27 Reference 1

=m AE. 30

1

= 1 + ex + 22x AE. 35

1

=1+ rx + 62X + c3x f l 4X + ,5x

= logh (1 + xl

1+x R. 425

= logh(x+ w) where-l dx< 1 L. 78

= logh (1 + m) where x2 < 1

AE. 43

= X

logh + -x

= 2/l + x2 logh {x + 1/l + x2}

A. 190

A. 191 T. 123

logh {x + w} where x < 1

= ; {logh (x + -)I2

A. 197

L. 77

= logh (1 - x -t x2) L. 79

= ex + logh (1 - x) where x c 1 F. 197

(38)

Series No.

(146) logh 2 + ; + ; - $ - . . . 00 (147)f ; - Bl(22 - 1)s x2 - I$(24 - 1,;

- B3(26- ‘)a X6 -... a

(148)t 2(8,(22 - 1) 5 - &(24 - 1) $+ Bj(26 - 1) $ - . . . m]

x2 x3

(149) e2 + m + r4 + . . . 00 x5 x9

(150) x + 3 + gj + . . . 00

(151)j logh 2 + logh 3 + . . . logh (n - 1) + ; logh n (This series is not convergent.)

BI* B3* W

(152)# 1 - ; + 2! x2 - 4! x4 + 6! x6 + . . . co

1 1

(153) 1 +q-Tzx2+Bx3 -goti+... 00

(154) 1 + g - -& + g7 +. . . 00 = c0 xy- l)n-1

‘+&2tZ+1)

(155) x3+++;+... 00 . ‘I

c+ p-4 CyGL?

(156) 5 - $‘I&, + .+,P,& -... a t For &, Bz, etc., see No. (1129).

$ For values of B1 and Bl*, etc., see No. (1129).

(39)

Reference

= logh (1 + cx) H. 498

A

=-

l x + 1 l x - 1

=- l x + 1

= 1 + *logh(l - x) x

= d (ex - e-x - j& + j&x) where j = 4-1

/&=;logh(2n)+nloghn-n s<b +--

4 4 (- l)l-1Br

+ 1.2.n -a- 3.4.n3 +**.+ (2r ^- 1).2pn2r-1 *.

=- X ex - I

= logh (; + x) Q .

= i tan-l x + i logh (1 + ^x2)

= logh (1 + X3@)

= [lo& (1 + XII’

r. I where ,Pk is the sum of all products k at a time, of the first ^rnatural numbers Y.80

N. 1543 N. 1544 F. 338 F. 338

1P. 612 22.123

W.243

.

Y.80

(40)

Series No.

x2 x4

(157) ; - z;i + m. -. . . 00

(158) x + ; + ; + $ - 7 + f + . . . 00 (159) Reversion of Series.

Y = x - blX2 - bgc3 - bp? -. . . ccl cm become

x = Y + GY2 + c2y3 + c3y4 + . . . 09

if Cl

c2 c3 c4 c5 c6

c7

See Van Orstrand (Phil. Msg. 19: 366.1910) for coefficients up to C,,.

(la) ’ - 2(n: 1) - 2.3(n1+ 1)2 - 3$+ ¹⁾³ ^-“- ^O”

(161) 1 + ; + ; + $ + . . . co

. . .

(162) $ + & + k6 + . . . co (163) ; + $2 + k3 + . . . 00

(41)

= logh 3 Y. 107

= logh (1 + x + x2 + x3 + x4) Y. 107

T. 116

= ₆₁

= b2 + 26,*

= 63 + %I& + 561~

= 64 + 66163 + 3622 + 21b1*62 + 14614

= 65 + 7(blb4 + b263) + 28(61*63 + blb2*) + 8461362 + 4261’

= b6 + 4(2b,b5 + 2b2b4 + b3*) + 12(3b,*b4 + 6b,b2b3 + b23 + 60(2b13b3 + 3b1*b2*) + 330b,4b2 + 132b16

= b, + 9(blb6 + b2b5 + b,bc,)

+ 45(bt2b5 + b,b3* + b2*b3 + 2b,b2b4) + 165(b13b4 + b,b23 + 3b1*b2b3)

+ 495(6,4b, + 2b13b2*) + 1287b,5b2 + 4296,’

= logh 5

F. 197 F. 197 F. 197 C. 368

(42)

Series No.

(164)

IX. Binomials. See also No. (1102).

(165) ^X” ⁺ ^np-‘a ⁺ ^!ipp-2n2 ⁺ . . . + @

(166) 1 +;xz+;x4+... 00

(167) 2 + &, + $ + E + . . . 00 . . * .

(168) 1 -f.;+~4.&~.&+... CQ . .

(169) 1 + ; + g + -&&$ . . + . . . co

1 1.3

(170) 1 +2x-r4 1.305

x2--x3+&g&4+... co

2.4.6

(171) 1 -~x-~4x2+~~3+fix4-... co

. . .

(The above two series are useful in forming certain trigonometrical series.)

(172) f x + g 22x2 + g 32x3 + . . . 00 . .

(173) 1 - x + x2 - x3 +. . . co (174) 1 - 2x + 3x2 - 4x3 + . . . OS

(175) 1 +zx-r4x2+y-J-Jx3 1 1.1 l-1.3 1.1.3.5

--x4+... UJ

(176) 1 - 1 + c4x2 zx 1.3 - -x3 2.406 l-3.5 + -x4 2.4.6.8 1.3.5.7 - *** co

(43)

= logh {logh (1 + x)1/x}

= (x + a)”

=&2

= 32/3

= q3

= z/8

= -+x J

1 + x*

x(x + 3)

= 9(1 - x)7/3

= (1 + x)-l

= (1 + x)-2

=diTi

BINOMIALS 33

Y. 107

H. 468 F. 167 F. 168 F. 168

T. 117 T. 117 T. 117 T. 117

(44)

Series No.

(177) 1 + 1 1.2.5.8

jx - =x2 1.2 + -x3 1.2.5 -

3.6.9 3.(j.g.12x4 +-” * (178) 1 - 5x + 1.4 x2 I c6 - 1.4.7 x3 mg + 1.4.7.10x4 3.6.9.12 _ ‘*’ O”

3 3.1 3.1.1 3.1.1.3

(179) 1 +zx+~4x~-~x~+-x4-... co

2.4.6.8

3 3.5 3.5.7

(180) 1 -zx+c4x2-mx3+... co

1 3 7 77

(181) 1 +4x-32x2+128x3-2o48x4+... co

(182) 1 -4x+Bx2-128x~+ 1 5 15 =x4 195 -... co

(183) 1 +;x-&x~+&~x~-$~x~+... co (184) 1 - ; x + & x2 - g5 x3 + -$ x4 - . . . co

(185) 1 + 1

gx - %x2 5 +-x3 55

1296 -- 311o4x4 935 + . . . co (186) 1 - ; x + &x2 - g6 x3 + g4 x4 - . . . co (187) 1 + n f + w(f)’

0

+ n(n - 4)(n - 5) x 3

3! 0 2 +... co (188) 1 +p2+ n2 nqn2 - 22) x4 + nyn2 - 22)(n2 - 42) xa+ 41

6! . . .

+;x+ n(n2 - 12)

3! x3+ n(n2 - 12)(n2 - 32)

5! x5 +... co

(45)

BINOMIALS

= (1 + x)1/3

= (1 + x)-1/3

= (1 + ,)3’Z

= (1 + x)-3/2

= (1 + x)1/4

35

Reference

T. 117 T. 117 T. 117 T. 117 T. 117

= (1 + x)-l/4 T. 117

= (1 + x)1/5 T. 118

= (1 + x)-l/S T. 118

= (1 + x)1/6 T. 118

= (1 + x)-1/6 T. 118

= & (1 + 2/l + x>” where x2 c 1 and n is any real number T. 118

= {x + 41 + ~2)” where x2 < 1 T. 118

(46)

Series No.

(189) c = m(m - *m - a...@ -n + 1)

m ” n!

(190) 1 + mCpx + . . . mCnxm + . . . ,c&P

(191) n&” + rnlG~rn& + m,Cz*m,G-2 + “,,C”

(192) 1 + ,c1 + ,c, + . . . ,c, + . . . (193) 1 - ,Cl + ,c, . . . (-l)n& +...

(194) l-&1 + 2.,C2x + . . . ne,C,xm-l + . . . (195) m(m - 1) + m(m - :{(m - 2, + . . .

+ m(m - 1). . .(m - r + 1) +

(r-2)! *-*

(1g6) 1 + m + m”,: 1) +. . . + m(m + *).,tm + r - 1)

(197) 2n

{ X” + fi2 x”-y + n(n - 3)

2!24 xn-4y4 + . . . + n(n - r - l)(n - r - 2). . . (n - 2r + 1) x~-2ry2,+

r!22r . . .

1 (198) 2$x2 + yz)llz

x 1 xn-’ + n-2

w x”-3y2 + (n - 3(n - 4) x”-5y4 +

2!24 . . .

+ (n - r - *)(n - r - 2). . .(n - 24 xn-2r-1y2r +

r!22r . . .

(199) 1 - ,Cl + ,c, - . . .( - l)n,C, (200) &1 -;*c2 + +-...

(47)

BINOMIALS

= (1 + x)m

= ^ml+q C n

= 2m where m > - 1

= 0 where m is positive

= m(1 + x)m-l

= m(m - 1)2m-2 where m 4: 1

= Cm + r>!

m!r! where r is a positive integer

37

Reference

c. 186 c. 186 C. 189 c. 191 c. 191 c. 197

c. 200 c. 200

= (x + dx2 + ~2)” + {x - z/x2 + y2>” where n is a positive integer C. 204

= {x + mp - {x - 4x2 + y2>” where n is a positive integer c. 205

= (- lp,,,-,C,, where n is a positive integer c. 210

=1+;+...: c. 212

(48)

Series No.

X. Simple Inverse Products

(201)t To find the sum of IZ terms of a series, each term of which is composed of the reciprocal of the product of r factors in arithmetical progression, the first factors of the several terms being in the same arithmetical progression:

Write down the nth term, strike off a factor from the beginning, divide by the number of factors so diminished, and by the common difference change the sign and add a constant.

1 1

- -

(202) 1.2.3.4 + 2.3.4.5 +“’ nterms (203) --- 3 + 4 + 3.4.6+...nterms 5

1.2.4 2.3.5

1 1 1

(204)r2+c3+r4+....nterms

+ . . . co

1 1 1

(205) m + n + r7 + . . . n terms 1

t206) m + 3.4 2.2 + ;5.22 +... n terms .

2 3

(207) 1a3e4 -t 2.4e5 +... n terms --

1 1 1

(208) r4 + c7 + no +- . . . n terms m

+ . . . co = c , (3n - 2;(3n + 1)

t In some cases the nth term can by partial fractions be resolved into the standard form when this rule can apply.

(49)

SIMPLE INVERSE PRODUCTS 39 Reference

F. 316

1 1

=-- 18 3(n + l)(n + 2)(n + 3) F. 317

29 --- 1 3 4

=% n+3 2(n+2)(n+3)-3(n+I)(n+2)(n+3) F*317

=- n

n+l F. 322

= 1

2n 1

=---

n+2 2

17 6n2 + 21n + 17

= % - 6(n + l)(n + 2

3(” + 3)

;

=- n

3n + 1

=- 1

3 ---o K@j&,$& = L] F-322

\

(50)

Series No.

(*w -- 1 1 1

1.3.5 +

3.5.7

+ 5a7.9 +... n terms + . . . co

1 1

(210)

-L

_1.4.7

+ -

^4.7.10

+ -

^7-10.13

+*** n terms + . . . m

(211)

A- + -L + -L +“’ nterms

^1.2.3 ^2.3.4 ^3.4.5

+ . . . a (l) --- 1 +

^3.4.5 ^4.5.6²

+

^5.6.7³

+*-’ nterms

+ . . . ca

(213)

-!- + 3

^1.2.3 ^2.3.4

+ -? +“- nterms

^3.4.5

+ . . . 00

(*14) ^r2.231 +

-.-

^2.3⁴¹ ²²

+ -.-

^{3.4 23}⁵¹ ^+... ^nterms

(215) Jf-.4+ 2.3 g4.42 .

+

^$43

.

^{+. . .}^{n terms}

(216) 1

+

² ³

m m +

^1.3.5.7

+*-. n terms

l-2 2-22 3.23

(217) 3! + 4! + 5! + . . . n terms (218) 1.2 2.3

3+

²⁺ ^3’4+...³³ ^Co

(219) +*.f f c3.; + ;4.$ +. . . n terms

(51)

SIMPLE INVERSE PRODUCTS

1 1

= i2 - 4(2n + 1)(2n + 3)

=T2 1

41

Reference

F. 322

1 1

= % - 6(3n + 1)(3n + 4)

=zl 1 F. 322

5 2n+5

= 5 - 2(n + l)(n + 2)

=- 5 4 1

= 6 - 23 + (n + 32x, + 4)

=- 1 6

3 2 1

= s - n+2 + 2(n + l)(n + 2)

=- 3 4

n - lb+’ 2

=-.-+-

n+2 3 3

1 1 1

= 2 --- 2 1.3-5.7.. .(2n + 1)

= l-&y

=- 9 4

=1--L.!.

n+l3n

F. 322

F. 322 F. 333 F. 333 F. 333 F. 333 F. 332 F. 331

(52)

Series No.

(220)

W)

(222)

(223)

;+A+&+...

_{. .} ^{n terms}

& + g4 + +j +... -

n(n + 2)

1 n

c

* (1 + nx)(l + n + lx) ¹ 1

(x + 1)(x + 2) + (x + 1)(x :! 2)(x + 3)

+ (x + 1)(x + 2’,;x + 3)(x + 4) + - * * co (224)

(225)

c co

1 (x + n)(xl+ n + 1) jtg

w c,

;+-‘-

^X2

u(u + 1) + u(u + l)(fz + 2) + * - * O”

a(u + 1) a(u + I>(u + 2)

(226) ii + b(b + I) x + b(b + l)(b + 2) x2 + * * * O”

(227) 1 + 1 .- 1 l-3 1 2n + 2 5 2n + 4

+

m’2n + 6

1.3-5 1

+ 5i7j.m +“’ C0

u(u + 1) a(u + l>(u + 2)

(228) l + t!t + b(b + 1) + b(b + l)(b + 2) + - - * co .

(53)

SIMPLE INVERSE PRODUCTS 43

1 1 1

= ---

2 2 3.7.11.. .(4n - 1) 36 + 5n

= 4(n + I)(n + 2) -

3Lj

- /

Reference

F. 331 w. 57

= (1 + x)(1 1 n + lx}

=- 1 x+1

Iz f- pij/MQ.)” &

3-j A

= (a - l)! x4 {6x - zs} whereaispositive T. 118

positive and a < b;

efficients

, etc., are binomial co- T. 118

2.4.6.. .2n

= 3.5.7.. .(2tl + 1) 1Z. 267

=b-a-l b-l where b - l>a>O A. 48

(54)

Series No.

(230) -!- x(x+ l)+

1 x(x + 1)(x + 2)

1.2

+ x(x + 1)(x + 2)(x + 3) + * * . Oc) (231) (1 + x)il + 2x) + (1 + 2x:(1 + 3x) + * * * n terms

1

(232) (1 + x)(1 + ax) + (1 + ax;;1 + &X) + * * * n terms XI. Other Inverse Products

A + & + . . . uz

1 1 1 1

+ r4 + r6 + n + . . . co

1 1 1

m + r4 - c5 + . . . ~0

1 1

- -

I

+ 3.4.5 + 5.6.7 +*-a O”

(237) 1 - -!- ’

l-2.3 3.4.5 + m - *** * I

<

(238) 1.3.5 + 1 5.7.9 + - 9.11.13 1 +*-* O” ,r I (23g) - 1 1.3.5 + - 1 3.5.7 + - 1 5.7.9 +**- co

(240) --- 1 1.3.5 3.5.7 1 + m 1 -*a* a3

(241) (&)2 + (&J2 + (A)1 + - - - O”

Summation of Series Dover Jolley