An Introduction to
The Harmonic Series
And Logarithmic Integrals
For High School Students
Up To Researchers
Ali Shadhar Olaikhan
Copyright © 2021 by Ali Shadhar Olaikhan.
All rights reserved. No part of this book may be reproduced, distributed, or transmitted in any form or by any means, including photocopying, recording, or other
electronic or mechanical methods, without the prior written permission of the publisher, except as provided by United States of America copyright law. For permission
requests, e-mail the publisher at
First edition published April 2021
Book cover designed by Islam Farid and Aqil Almosawi
LATEX class prepared by Elio A. Farina
ISBN 978-1-7367360-1-2 (eBook)
To my parents
Preface
The harmonic series and logarithmic integrals, which are strongly interrelated, are
not commonly found in the standard textbooks. Evaluating them can be challenging
to new learners, as it requires specific approaches and a good knowledge of special
functions such as the gamma function, the polygamma function, the beta function,
the polylogarithm function, and various other special functions and constants. It
also requires a lot of experience and patience, since it involves plenty of tricks and
time-consuming calculations.
The purpose of this book is to introduce the harmonic series in a way suitable for all
readers with a good knowledge of calculus, from high school students to researchers.
The book is the result of over five years of working on the harmonic series. As
I taught myself this topic, I struggled to find the proofs for most of the identities
required for evaluating the harmonic series. With the experience gained over years,
I managed to prove these identities in detail using only basic definitions and well–
known techniques, and without using contour integration or the residue theorem,
which require a deep understanding of complex analysis.
I would like to inform the reader that I borrowed a few proofs from some sites, mainly
from the Mathematics Stack Exchange site, adding more details and modifying them
my own way. Also, most of the text is written in equations, so the reader won’t find
much unnecessary verbiage in this book.
The book consists of four chapters. Chapter 1 presents some essential series transformations and special functions and shows how these functions are related to each
other. It explains the definition and properties of each function and also derives many
special values needed for the calculations in chapters 3 and 4.
In chapter 2, the reader will find the derivations of plenty of useful identities: generating functions involving the harmonic number and series expansion of powers of
arcsin(x). Other identities are derived using the beta function, the Cauchy product,
Abel’s summation, and Fourier series.
Chapter 3 prepares all the integral results required to calculate the harmonic series in
chapter 4, including some new results. These were derived using algebraic identities,
integral manipulations and the beta function.
Chapter 4 shows how to calculate many types of harmonic series: non-alternating
series, alternating series, series with powers of 2 in the denominator, series with
powers of 2n + 1 in the denominator, series with rational argument, series with skew
iv
Preface
v
harmonic number, series with central binomial coefficient, and many others. Several
solutions are presented using two different methods.
At the end of the book, I have provided a table of Mathematica commands for
approximating or evaluating limits, derivatives, integrals, and series, so that the
reader can verify any result of interest throughout the book.
More advanced and challenging problems about the harmonic series may be found
on my Mathematics Stack Exchange page,
/users/432085/ali-shadhar. I decided not to include them in the book for the
sake of simplicity. To keep up to date with any new identities or results, you can
follow my Facebook group, Harmonic Series, />ups/178723409566339.
Finally, I would like to express my gratitude to my friend Cornel Ioan Vălean for
being a big motivation for me to explore the realm of the harmonic series through his
amazing problems and solutions, many of which are included in his book, (Almost)
Impossible Integrals, Sums, and Series, and for his valuable tips for writing this book.
I would also like to thank Elio Arturo and my brother Hasan Shadhar for their help in
using LaTeX. I extend my gratitude to my friends, Khalaf Ruhemi, Shivam Sharma,
and Hasan Hussein for all the support and encouragement they offered me while
writing this book. I also want to thank my parents, to whom I am dedicating this
book, for all their support.
Phoenix, Arizona, USA
April 2021
Ali Shadhar Olaikhan
Contents
1
Series Transformations and Special Functions
1.1 Shifting the Sum Index . . . . . . . . . . . . . .
1.2 Reversing the Order of the Sum Terms . . . . . .
1.3 Splitting a Sum Into its Odd and Even Parts . . .
1.4 Converting the Summand a2n to an . . . . . . .
1.5 Converting the Summand a2n+1 to an . . . . . .
1.6 Converting the Summand (−1)n a2n to in an . . .
1.7 Converting the Summand (−1)n a2n+1 to in an .
1.8 Converting a Sum to a Product . . . . . . . . . .
1.9 Double Summation . . . . . . . . . . . . . . . .
1.10 The Logarithm of a Complex Number . . . . . .
1.11 Gamma Function . . . . . . . . . . . . . . . . .
1.11.1 Definition . . . . . . . . . . . . . . . . .
1.11.2 Functional Equation . . . . . . . . . . .
1.11.3 Stirling’s Approximation . . . . . . . . .
1.11.4 Expressing Gamma Function as a Product
1.11.5 Euler’s Definition as an Infinite Product .
1.11.6 Euler’s Reflection Formula . . . . . . . .
1.11.7 Legendre Duplication Formula . . . . . .
1.12 Beta Function . . . . . . . . . . . . . . . . . . .
1.12.1 Definition . . . . . . . . . . . . . . . . .
1.12.2 Trigonometric Integral Representation . .
1.12.3 Improper Integral Representation . . . .
1.12.4 Powerful Integral Representation . . . . .
1.13 Riemann Zeta Function . . . . . . . . . . . . . .
1.13.1 Definition . . . . . . . . . . . . . . . . .
1.13.2 Integral Representation . . . . . . . . . .
1.13.3 Evaluation of ζ(0) . . . . . . . . . . . .
1.13.4 Evaluation of ζ(2) . . . . . . . . . . . .
1.13.5 Evaluation of ζ(2n) . . . . . . . . . . .
1.14 Dirichlet Eta Function . . . . . . . . . . . . . . .
1.14.1 Definition . . . . . . . . . . . . . . . . .
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1
1
2
3
4
4
5
6
6
7
8
10
10
10
11
12
13
14
16
17
17
18
18
18
19
19
19
20
21
22
25
25
vi
Contents
2
vii
1.14.2 Integral Representation . . . . . . . . . . . . . . .
1.15 Dirichlet Beta Function . . . . . . . . . . . . . . . . . . .
1.15.1 Definition . . . . . . . . . . . . . . . . . . . . . .
1.15.2 Integral Representation . . . . . . . . . . . . . . .
1.15.3 Evaluation of β(2a) . . . . . . . . . . . . . . . .
1.15.4 Evaluation of β(2a + 1) . . . . . . . . . . . . . .
1.16 Polylogarithm Function . . . . . . . . . . . . . . . . . . .
1.16.1 Definition . . . . . . . . . . . . . . . . . . . . . .
1.16.2 Dilogarithm Reflection Formula . . . . . . . . . .
1.16.3 Landen’s Dilogarithm Identity . . . . . . . . . . .
1.16.4 Dilogarithm Inversion Formula . . . . . . . . . . .
1.16.5 Relation Involving Four Dilogarithm Functions . .
1.16.6 Another Relation Involving Dilogarithm Functions
1.16.7 Landen’s Trilogarithm Identity . . . . . . . . . . .
1.16.8 Polylogarithm Inversion Formula . . . . . . . . . .
1.17 Harmonic Number . . . . . . . . . . . . . . . . . . . . .
1.17.1 Definition . . . . . . . . . . . . . . . . . . . . . .
1.17.2 Infinite Series Representation . . . . . . . . . . .
1.18 Skew Harmonic Number . . . . . . . . . . . . . . . . . .
1.18.1 Definition . . . . . . . . . . . . . . . . . . . . . .
1.18.2 Infinite Series Representation . . . . . . . . . . .
1.19 Digamma Function . . . . . . . . . . . . . . . . . . . . .
1.19.1 Definition . . . . . . . . . . . . . . . . . . . . . .
1.19.2 Digamma Reflection Formula . . . . . . . . . . .
1.19.3 Digamma–Harmonic Number Identity . . . . . . .
1.20 Polygamma Function . . . . . . . . . . . . . . . . . . . .
1.20.1 Definition . . . . . . . . . . . . . . . . . . . . . .
1.20.2 Series Representation . . . . . . . . . . . . . . . .
1.20.3 Integral Representation . . . . . . . . . . . . . . .
1.20.4 Evaluation of ψ (a) (1) . . . . . . . . . . . . . . . .
1.20.5 Evaluation of ψ (a) 12 . . . . . . . . . . . . . . .
1.20.6 Evaluation of ψ (2a) 14
. . . . . . . . . . . . . .
1.20.7 Evaluation of ψ (2a) 34
. . . . . . . . . . . . . .
1.21 Catalan’s Constant . . . . . . . . . . . . . . . . . . . . .
1.22 Euler–Mascheroni Constant . . . . . . . . . . . . . . . . .
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26
26
26
27
27
29
30
30
34
35
36
36
37
37
39
40
40
43
44
44
46
47
47
47
48
50
50
50
51
52
52
53
53
54
55
Generating Functions and Powerful Identities
2.1 Generating Functions . . . . . . . . . . . .
(a) n
∞
. . . . . . . . . . .
2.1.1
n=1 Hn x
∞ Hn n
2.1.2
x
.
.
. . . . . . . . . .
n=1 n
∞ Hn n
2.1.3
. . . . . . . . . . . .
n=1 n2 x
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56
57
57
59
59
. . . . . . . . . . . . . . . . . . . .
60
. . . . . . . . . . . . . . . . . . . .
61
2.1.4
2.1.5
(2)
∞ Hn
n
. . . .
n=1 n x
(2) n
∞
2
n=1 (Hn − Hn )x
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viii
Contents
2.1.6
2.1.7
2.1.8
2.1.9
2.1.10
2.1.11
2.1.12
2.1.13
2.1.14
2.1.15
2.1.16
2.1.17
2.1.18
2.1.19
2.1.20
2.1.21
2.1.22
2.1.23
2.1.24
2.1.25
2.1.26
2.1.27
2.1.28
2.2
2.3
2.4
2
(2)
−Hn
∞ Hn
xn . .
n=1
n
2
∞ Hn n
. . . . .
n=1 n x
∞ Hn n
. . . . .
n=1 n3 x
(2)
∞ Hn
n
. . . .
n=1 n2 x
(3)
∞ Hn
n
. . . .
n=1 n x
∞
3 n
H
x
.
.
. . .
n=1 n
2
∞ Hn
n
. . . . .
n=1 n2 x
(2) n
∞
. .
n=1 Hn Hn x
(2)
∞
3
−
3H
H
(H
n n
n
n=1
(2)
∞ Hn Hn
xn . . .
n=1
n
3
∞ Hn n
. . . . .
n=1 n x
62
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
63
63
. . . . . . . . . . . . . . . . . . . .
67
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
67
68
. . . . . . . . . . . . . . . . . . . .
68
. . . . . . . . . . . . . . . . . . . .
(3)
+ 2Hn )xn . . . . . . . . . . . . .
69
70
. . . . . . . . . . . . . . . . . . . .
70
H n xn
Hn n
n x
. . . . . . . . . . . . . . . . . . .
(3)
(2)
(4)
+ 8Hn Hn + 3(Hn )2 − 6Hn )xn .
. . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . .
.
.
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.
72
72
73
74
Hn n
n2 x
n n
. . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . .
75
78
. . . . . . . . . . . . . . . . . . . . . . . .
79
. . . . . . . . . . . . . . . . . . . . . . . .
80
∞
4
n=1 (Hn
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
. . . . . . . . . . . . . . . . . . . .
−
(2)
6Hn2 Hn
H2x
Hn/2 n
n x
Hn/2 n
n2 x
(2n
n)
n
4n Hn x . .
(2n
n ) Hn n
4n n x . .
2n
( n ) Hn n
4n n2 x . .
2H2n −Hn 2n
x
n
H2n 2n+1
.
2n+1 x
. . . . . . . . . . . . . . . . . . . . .
82
. . . . . . . . . . . . . . . . . . . . .
83
. . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . .
84
85
86
n
(−1) H2n 2n+1
. . . . .
2.1.29
2n+1 x
Hn −H2n
1
2.1.30
− 2n2 x2n . .
n
Series Expansion of Powers of arcsin(z)
2.2.1 Series Expansion of arcsin(z) .
√
2.2.2 Series Expansion of arcsin(z)
. .
1−z 2
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87
88
89
89
90
2.2.3 Series Expansion of arcsin3 (z) . . . . .
2.2.4 Series Expansion of arcsin4 (z) . . . . .
Identities by Beta Function . . . . . . . . . . . .
2.3.1 Expressing Beta Function as a Product . .
2.3.2 Evaluation of Four Logarithmic Integrals
Identities by Cauchy Product . . . . . . . . . . .
2.4.1 Cauchy Product of Two Power Series . .
2.4.2 Cauchy Product of − ln(1 − x) Li2 (x) . .
2.4.3 Cauchy Product of Li22 (x) . . . . . . . .
2.4.4 Cauchy Product of − ln(1 − x) Li3 (x) . .
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. 91
. 92
. 95
. 95
. 95
. 99
. 99
. 99
. 100
. 101
Contents
2.5
2.6
3
ix
2.4.5 Cauchy Product of Li2 (x) Li3 (x) . . .
2.4.6 Cauchy Product of Li23 (x) . . . . . . .
2.4.7 Cauchy Product of − ln(1 − x) Li4 (x) .
Identities by Abel’s Summation . . . . . . . . .
2.5.1 Abel’s Summation . . . . . . . . . . .
2.5.2 First Application . . . . . . . . . . . .
2.5.3 Second Application . . . . . . . . . . .
2.5.4 Third Application . . . . . . . . . . . .
Identities By Fourier Series . . . . . . . . . . .
2.6.1 Fourier Series . . . . . . . . . . . . . .
2.6.2 Fourier Series of Even Function . . . .
2.6.3 Fourier Series of Odd Function . . . .
2.6.4 Fourier Series of cos(zx) . . . . . . . .
2.6.5 Fourier Series of sin(zx) . . . . . . . .
2.6.6 Fourier Series of ln(sin x) . . . . . . .
2.6.7 Fourier Series of ln(cos x) . . . . . . .
2.6.8 Fourier Series of ln(tan x) . . . . . . .
π
2.6.9 Series Representation of sin(πz)
. . . .
2.6.10 Series Representation of cot(πz) . . . .
2.6.11 Euler’s Product Formula of sin(πz) . .
2.6.12 Series Representation of sec π2 z . . .
2.6.13 Series Representation of sin(x) . . . .
2.6.14 Series Representation of tan x ln(sin x)
2.6.15 Series Representation of ln2 (2 cos x) .
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102
103
104
105
105
107
108
110
112
112
114
115
116
117
118
121
122
122
122
123
124
125
125
126
Logarithmic Integrals
128
3.1 Generalized Logarithmic Integrals . . . . . . . . . . . . . . . . . . 128
1 lna (x)
3.1.1
dx . . . . . . . . . . . . . . . . . . . . . . . . . . 128
0 1−x
3.1.2
3.1.3
3.1.4
3.1.5
3.1.6
3.1.7
3.1.8
3.1.9
3.1.10
3.1.11
3.1.12
3.1.13
lna (x)
1+x dx . . . . . .
lna ( 1−x
1+x )
dx . . . .
x
1−x
ln( 1+x ) lna−1 (x)
dx
x
lna (1−x)
dx
.
.
. .
1+x
1
a
ln
(x)
2
1−x dx . . . . .
0
1 lna (1+x)
dx . . . .
x
0
2a−1
x
( 1−x
)
1 ln
dx . .
1+x
0
∞ lna (1+x)
1+x2 dx . . . .
0
1 lna (1−x)
1+x2 dx . . . .
0
∞ ln2a (x)
1+x2 dx . . . . .
0
∞ Lia (−x)
1+x2 dx . . . .
0
1 Li2a+1 (−x)
dx . . .
1+x2
0
1
0
1
0
1
0
1
0
. . . . . . . . . . . . . . . . . . . . 129
. . . . . . . . . . . . . . . . . . . . 130
. . . . . . . . . . . . . . . . . . . . 131
. . . . . . . . . . . . . . . . . . . . 131
. . . . . . . . . . . . . . . . . . . . 132
. . . . . . . . . . . . . . . . . . . . 134
. . . . . . . . . . . . . . . . . . . . 135
. . . . . . . . . . . . . . . . . . . . 136
. . . . . . . . . . . . . . . . . . . . 137
. . . . . . . . . . . . . . . . . . . . 137
. . . . . . . . . . . . . . . . . . . . 139
. . . . . . . . . . . . . . . . . . . . 140
x
Contents
3.1.14
3.1.15
3.1.16
3.1.17
3.1.18
3.1.19
3.1.20
3.1.21
3.1.22
3.1.23
3.1.24
3.1.25
3.1.26
3.1.27
3.1.28
3.1.29
3.1.30
3.1.31
3.2
1 ln2a (x) ln(1+x)
dx . .
1+x2
0
∞ Lia (−x2 )
1+x2 dx . . . . .
0
1 Li2a+1 (−x2 )
dx . . . .
1+x2
0
1 ln2a (x) arctan(x)
dx . .
1−x2
0
∞ ln2a (x) ln(1+x)
√
dx . .
0
x(1+x)
1 ln2a (x) ln(1+x2 )
dx . .
1+x2
0
1 lna (x) lna (1−x)
dx . .
x(1−x)
0
1
a
a
ln
(x)
ln
(1−x)
2
dx . .
x(1−x)
0
1 lna (x) ln(1−x)
dx . . .
1−x
0
1 lna (x) ln(1−x)
dx . . .
x(1−x)
0
1 lna (x) ln(1+x)
dx . . .
1+x
0
1 lna (x) ln(1+x)
dx . . .
x(1+x)
0
1 lna (1−x) ln(1+x)
dx .
x
0
1+x
a
1 ln (x) ln( 2 )
dx . . .
1−x
0
1 lna (1−x) Li2 (x)
dx . .
x
0
∞ ln2a−1 (x) ln(1+x)
dx .
x(1+x)
0
1 x2n
dx . . . . . . . .
0 1+x
1 xn
dx . . . . . . . .
0 1+x
1 2n−1
x
arctanh(x)dx
0
1 n−1
x
Li
a (x)dx . . .
0
3.1.32
3.1.33
Results of Logarithmic Integrals
π
2
3.2.1
sin(2nx) cot(x)dx .
0
3.2.2
3.2.3
3.2.4
3.2.5
3.2.6
3.2.7
3.2.8
3.2.9
3.2.10
3.2.11
3.2.12
3.2.13
π
2
0π
2
0π
2
. . . . . . . . . . . . . . . . . . . 141
. . . . . . . . . . . . . . . . . . . 142
. . . . . . . . . . . . . . . . . . . 143
. . . . . . . . . . . . . . . . . . . 144
. . . . . . . . . . . . . . . . . . . 145
. . . . . . . . . . . . . . . . . . . 147
. . . . . . . . . . . . . . . . . . . 147
. . . . . . . . . . . . . . . . . . . 148
. . . . . . . . . . . . . . . . . . . 149
. . . . . . . . . . . . . . . . . . . 149
. . . . . . . . . . . . . . . . . . . 150
. . . . . . . . . . . . . . . . . . . 150
. . . . . . . . . . . . . . . . . . . 151
. . . . . . . . . . . . . . . . . . . 151
. . . . . . . . . . . . . . . . . . . 152
. . . . . . . . . . . . . . . . . . . 153
. . . . . . . . . . . . . . . . . . . 154
. . . . . . . . . . . . . . . . . . . 155
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156
156
158
158
ln(sin x)dx . . . . . . . . . . . . . . . . . . . . . . . . 159
ln2 (sin x)dx . . . . . . . . . . . . . . . . . . . . . . . 161
ln(sin x) ln(cos x)dx . . . . . . . . . . . . . . . . . . . 162
. . . . . . . . . . . . . . . . . . . 163
0
1 ln(x) ln(1−x)
√
dx . . .
0
x 1−x
∞ ln2 (x) ln(1+x2 )
dx . .
1+x2
0
1 ln(1−x) ln(1+x)
dx . .
x
0
1 ln(x) ln(1−x) ln(1+x)
dx
x
0
1 ln(1−x) ln2 (1+x)
dx . .
x
0
1 ln2 (1−x) ln(1+x)
dx . .
x
0
1 ln3 (1−x) ln(1+x)
dx . .
x
0
1 ln(1−x) ln3 (1+x)
dx . .
x
0
1 ln3 (1+x) ln(x)
dx . . .
x
0
. . . . . . . . . . . . . . . . . . . 165
. . . . . . . . . . . . . . . . . . . 167
. . . . . . . . . . . . . . . . . . . 168
. . . . . . . . . . . . . . . . . . . 169
. . . . . . . . . . . . . . . . . . . 170
. . . . . . . . . . . . . . . . . . . 170
. . . . . . . . . . . . . . . . . . . 172
. . . . . . . . . . . . . . . . . . . 172
Contents
xi
3.2.16
1 ln(x) ln(1+x)
dx
1−x
0
1 ln(x) ln(1−x)
dx
1+x
0
1 ln(x) ln2 (1−x)
dx
1+x
0
3.2.17
1
1
1 ln2 (1+x)
ln(1+x)
ln(1+x)
dx, 0 ln(1−x)
dx, & 0 ln(x)1+x
dx
2
0
1+x2
1+x2
3.2.14
3.2.15
3.2.18
3.2.19
3.2.20
3.2.21
3.2.22
3.2.23
3.2.24
3.2.25
3.2.26
3.2.27
3.2.28
3.2.29
4
. . . . . . . . . . . . . . . . . . . . . . 174
. . . . . . . . . . . . . . . . . . . . . . 175
&
1 ln(x) ln2 (1−x)
√
dx .
0
x(1−x)
1
2
ln
(x)
ln(1−x)
2
dx .
1−x
0
1 ln2 (x) ln(1+x)
dx .
1+x
0
1 ln3 (1−x) ln(x)
dx .
1
x
2
1 Li2 (−x)
dx . . . .
0 1+x2
1 ln(x) arctan x
dx .
1+x
0
1 ln2 (x) arctan x
dx .
x(1+x2 )
0
1 Li22 (−x)
dx . . . .
x
0
1
2
Li
(−x)
2
2
dx . . . .
x
0
1 ln2 (1−x) Li2 (x)
dx
x
0
1 ln3 (1−x) Li2 (x)
dx
x
0
1 ln4 (1−x) Li2 (x)
dx
x
0
1 ln2 (x) ln(1−x)
dx
1+x
0
. . . . . . . . . . 175
. . . 177
. . . . . . . . . . . . . . . . . . . . . 179
. . . . . . . . . . . . . . . . . . . . . 180
. . . . . . . . . . . . . . . . . . . . . 181
. . . . . . . . . . . . . . . . . . . . . 181
. . . . . . . . . . . . . . . . . . . . . 183
. . . . . . . . . . . . . . . . . . . . . 183
. . . . . . . . . . . . . . . . . . . . . 184
. . . . . . . . . . . . . . . . . . . . . 186
. . . . . . . . . . . . . . . . . . . . . 187
. . . . . . . . . . . . . . . . . . . . . 188
. . . . . . . . . . . . . . . . . . . . . 188
. . . . . . . . . . . . . . . . . . . . . 188
Harmonic Series
4.1 Generalized Harmonic Series . . . . . . . . . . . . . . . . . . . . .
∞ Hn/p
4.1.1
n=1 nq . . . . . . . . . . . . . . . . . . . . . . . . . .
∞ Hn
4.1.2
n=1 nq . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.3
4.1.4
4.1.5
4.1.6
4.1.7
4.1.8
4.1.9
4.1.10
4.1.11
4.1.12
4.1.13
4.1.14
4.2
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
Hn
nq . . . . .
(−1)n Hn
. .
n2q
(−1)n H n
. .
n2q
Hn/2
.
.
. .
n2q
(−1)n Hn/2
.
n2q
(q)
ζ(q)−Hn
.
n
(2)
Hn
. . .
n2q+1
(2q+1)
Hn
. .
n2
2
Hn
. . .
n2q+1
Hn
. .
q
(2n+1)
(q)
(−1)n Hn
.
n
(2q+1)
(−1)n Hn
2n+1
(−1)n Hn
(2n+1)2q+1 .
189
189
189
190
. . . . . . . . . . . . . . . . . . . . . . 194
. . . . . . . . . . . . . . . . . . . . . . 198
. . . . . . . . . . . . . . . . . . . . . . 201
. . . . . . . . . . . . . . . . . . . . . . 202
. . . . . . . . . . . . . . . . . . . . . . 202
. . . . . . . . . . . . . . . . . . . . . . 203
. . . . . . . . . . . . . . . . . . . . . . 205
. . . . . . . . . . . . . . . . . . . . . . 207
. . . . . . . . . . . . . . . . . . . . . . 208
. . . . . . . . . . . . . . . . . . . . . . 210
. . . . . . . . . . . . . . . . . . . . . . 212
. . . . . . . . . . . . . . . . . . . . . 213
4.1.15
. . . . . . . . . . . . . . . . . . . . . . 214
Non–Alternating Harmonic Series . . . . . . . . . . . . . . . . . . 216
xii
Contents
4.2.1
4.2.2
4.2.3
4.2.4
4.2.5
4.2.6
4.2.7
4.2.8
4.2.9
4.2.10
4.2.11
4.2.12
4.2.13
4.2.14
4.2.15
4.2.16
4.2.17
4.2.18
4.2.19
4.2.20
4.2.21
4.2.22
4.2.23
4.3
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
Hn
n2 . .
(2)
Hn
.
n2
2
Hn
n2 . .
Hn H2n
n2
(2)
Hn
.
n3
(3)
Hn
.
n2
2
Hn
n3 . .
(2)
Hn Hn
n2
3
Hn
n2 . .
(2)
Hn
.
n4
2
Hn
n4 . .
(4)
Hn
.
n2
(2) 2
(Hn
)
n2
(3)
Hn Hn
n2
2
(2)
Hn
Hn
n2
4
Hn
n2 . .
(2)
Hn Hn
n3
3
Hn
n3 . .
(2)
Hn
.
n5
2
Hn
n5 . .
(3)
Hn
.
n4
(4)
Hn
.
n3
2
(2)
Hn
Hn
n3
(2)
Hn
.
n7
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
(2)
(−1)n Hn
n
(3)
(−1)n Hn
n
n
(−1) Hn
.
n3
(2)
(−1)n Hn
n2
2
(−1)n Hn
.
n2
. . . . . . . . . . . . . . . . . . . . . . . . . 216
. . . . . . . . . . . . . . . . . . . . . . . . . 216
. . . . . . . . . . . . . . . . . . . . . . . . . 217
. . . . . . . . . . . . . . . . . . . . . . . . . 218
. . . . . . . . . . . . . . . . . . . . . . . . . 220
. . . . . . . . . . . . . . . . . . . . . . . . . 220
. . . . . . . . . . . . . . . . . . . . . . . . . 220
. . . . . . . . . . . . . . . . . . . . . . . . 222
. . . . . . . . . . . . . . . . . . . . . . . . . 225
. . . . . . . . . . . . . . . . . . . . . . . . . 225
. . . . . . . . . . . . . . . . . . . . . . . . . 226
. . . . . . . . . . . . . . . . . . . . . . . . . 226
. . . . . . . . . . . . . . . . . . . . . . . . . 227
. . . . . . . . . . . . . . . . . . . . . . . . 227
. . . . . . . . . . . . . . . . . . . . . . . . 228
. . . . . . . . . . . . . . . . . . . . . . . . . 230
. . . . . . . . . . . . . . . . . . . . . . . . 230
. . . . . . . . . . . . . . . . . . . . . . . . . 231
. . . . . . . . . . . . . . . . . . . . . . . . . 232
. . . . . . . . . . . . . . . . . . . . . . . . . 233
. . . . . . . . . . . . . . . . . . . . . . . . . 234
. . . . . . . . . . . . . . . . . . . . . . . . . 234
. . . . . . . . . . . . . . . . . . . . . . . . 235
4.2.24
. . . .
Alternating Harmonic Series
∞ (−1)n Hn
4.3.1
. . .
n=1
n
∞ (−1)n H2n
4.3.2
. .
n=1
n
∞ (−1)n Hn
4.3.3
. . .
n=1
n2
∞ (−1)n H2n
4.3.4
. .
n=1
n2
4.3.5
4.3.6
4.3.7
4.3.8
4.3.9
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236
238
238
238
239
240
. . . . . . . . . . . . . . . . . . . . . . . 240
. . . . . . . . . . . . . . . . . . . . . . . 241
. . . . . . . . . . . . . . . . . . . . . . . 241
. . . . . . . . . . . . . . . . . . . . . . . 243
. . . . . . . . . . . . . . . . . . . . . . . 243
Contents
xiii
4.3.10
4.3.11
4.3.12
4.3.13
4.3.14
4.3.15
4.3.16
4.3.17
4.4
(2)
(−1)n Hn Hn
n
3
(−1)n Hn
. .
n
(−1)n Hn
. .
n4
(2)
(−1)n Hn
.
n3
2
(−1)n Hn
. .
n3
(4)
(−1)n Hn
.
n
(3)
(−1)n Hn
.
n2
(2)
(−1)n Hn Hn
n2
3
(−1)n Hn
. .
n2
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
(2)
Hn
n2n .
2
Hn
n2n .
Hn
n3 2n .
(2)
Hn
n2 2n .
2
Hn
n2 2n .
(3)
Hn
n2n .
Hn
n4 2n .
(4)
Hn
n2n .
(2)
Hn
n3 2n .
(3)
Hn
n2 2n .
2
Hn
n3 2n .
(2)
Hn Hn
n2 2n
3
Hn
n2 2n .
. . . . . . . . . . . . . . . . . . . . . . 244
. . . . . . . . . . . . . . . . . . . . . . 245
. . . . . . . . . . . . . . . . . . . . . . 245
. . . . . . . . . . . . . . . . . . . . . . 247
. . . . . . . . . . . . . . . . . . . . . . 247
. . . . . . . . . . . . . . . . . . . . . . 248
. . . . . . . . . . . . . . . . . . . . . . 249
. . . . . . . . . . . . . . . . . . . . . . 249
4.3.18
. . . . . . . . . . . . . . .
Harmonic Series with Powers of 2 in the Denominator
∞
Hn
4.4.1
n=1 n2n . . . . . . . . . . . . . . . . . . .
∞
Hn
4.4.2
n=1 n2 2n . . . . . . . . . . . . . . . . . . .
4.4.3
4.4.4
4.4.5
4.4.6
4.4.7
4.4.8
4.4.9
4.4.10
4.4.11
4.4.12
4.4.13
4.4.14
4.5
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
251
252
252
252
. . . . . . . . . . . . . . . . . . . . . . . . . 253
. . . . . . . . . . . . . . . . . . . . . . . . . 254
. . . . . . . . . . . . . . . . . . . . . . . . . 254
. . . . . . . . . . . . . . . . . . . . . . . . . 255
. . . . . . . . . . . . . . . . . . . . . . . . . 256
. . . . . . . . . . . . . . . . . . . . . . . . . 256
. . . . . . . . . . . . . . . . . . . . . . . . . 257
. . . . . . . . . . . . . . . . . . . . . . . . . 258
. . . . . . . . . . . . . . . . . . . . . . . . . 258
. . . . . . . . . . . . . . . . . . . . . . . . . 259
. . . . . . . . . . . . . . . . . . . . . . . . . 260
. . . . . . . . . . . . . . . . . . . . . . . . 262
4.4.15
. . . . . . . . . . . . . . . . . . . . . . . . . 265
Harmonic Series with
Powers
of 2n + 1 in the denominator . . . . . 266
∞ (−1)n H2n+1
.
. . . . . . . . . . . . . . . . . . . . . 266
4.5.1
n=0
2n+1
4.5.2
4.5.3
4.5.4
4.5.5
4.5.6
4.5.7
4.5.8
∞ (−1)n H2n+1
n=0
(2n+1)2
(2)
n
∞ (−1) H2n+1
n=0
2n+1
∞
Hn
n=1 (2n+1)2 . .
∞ (−1)n Hn
n=0 (2n+1)2 . .
(2)
∞ (−1)n Hn
.
n=0
2n+1
∞ (−1)n H2n+1
n=0
(2n+1)3
(2)
n
∞ (−1) H2n+1
n=0
(2n+1)2
. . . . . . . . . . . . . . . . . . . . . . 266
. . . . . . . . . . . . . . . . . . . . . . 267
. . . . . . . . . . . . . . . . . . . . . . 267
. . . . . . . . . . . . . . . . . . . . . . 268
. . . . . . . . . . . . . . . . . . . . . . 269
. . . . . . . . . . . . . . . . . . . . . . 271
. . . . . . . . . . . . . . . . . . . . . . 272
xiv
Contents
4.5.9
4.5.10
4.5.11
4.5.12
4.6
4.6.3
4.6.4
4.6.5
4.6.6
4.7.3
4.7.4
4.7.5
4.7.6
4.7.7
4.9
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
Hn
n3 . . . . .
(−1)n H n
. .
n3
(−1)n H n Hn
n
H n Hn
. . .
2
n
(−1)n H n Hn
n2
H 2n H2n
. .
n2
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
Hn/2
n2 . . . .
(−1)n Hn/2
.
n2
Hn/2
n3 . . . .
(−1)n Hn/2
.
n3
Hn Hn/2
. .
n2
(−1)n Hn Hn/2
n2
(−1)n Hn/2
.
n4
. . . . . . . . . . . . . . . . . . . . . . . . 272
. . . . . . . . . . . . . . . . . . . . . . . . 275
. . . . . . . . . . . . . . . . . . . . . . . . 275
. . . . . . . . . . . . . . . . . . . . . . . . 276
∞
(−1)n H (3)
. . . . . . . . . . . . . . . . . . . . . . 277
. . . . . . . . . . . . . . . . . . . . . . 278
. . . . . . . . . . . . . . . . . . . . . . 279
. . . . . . . . . . . . . . . . . . . . . . 280
. . . . . . . . . . . . . . . . . . . . . . 281
4.6.7
. . . . . . . . . . . . . . . . . . . . . . 284
Harmonic Series with Rational Argument . . . . . . . . . . . . . . 285
n
∞ (−1) Hn/2
4.7.1
. . . . . . . . . . . . . . . . . . . . . . . 285
n=1
n
4.7.2
4.8
(2)
Hn
(2n+1)2
2
Hn
(2n+1)2
(2)
Hn
(2n+1)3
(3)
Hn
(2n+1)2
(3)
Hn
(2n+1)3
n
+ 4 n=1
. . . . . . . . . . . . . 276
4.5.13
n3
Skew Harmonic Series
.
.
.
.
.
.
.
.
.
.
.
. . . . . . . . . . . . . 277
∞ (−1)n H n
4.6.1
. . . . . . . . . . . . . . . . . . . . . . . . 277
n=1
n
4.6.2
4.7
∞
n=1
∞
n=1
∞
n=1
∞
n=1
∞
n=1
. . . . . . . . . . . . . . . . . . . . . . 286
. . . . . . . . . . . . . . . . . . . . . . 286
. . . . . . . . . . . . . . . . . . . . . . 287
. . . . . . . . . . . . . . . . . . . . . . 288
. . . . . . . . . . . . . . . . . . . . . . 288
. . . . . . . . . . . . . . . . . . . . . 289
4.7.8
. . . . . . . . . . . . . . . . . . .
Harmonic Series2nwith Binomial Coefficient in the Numerator .
∞ ( n ) Hn
4.8.1
. . . . . . . . . . . . . . . . . . . . .
n=1 4n n
2n
∞ ( n ) (−1)n Hn
4.8.2
. . . . . . . . . . . . . . . . . .
n=1 4n
n
2n
(
)
∞
Hn
n
. . . . . . . . . . . . . . . . . . . . .
4.8.3
n=1 4n n2
2n
(2)
∞ ( n ) Hn
4.8.4
n=1 4n
n . . . . . . . . . . . . . . . . . . . . .
2n
(2)
∞ ( n ) H2n
4.8.5
n=1 4n
n . . . . . . . . . . . . . . . . . . . . .
2n
2
∞ ( n ) Hn
4.8.6
. . . . . . . . . . . . . . . . . . . . .
n=1 4n n
2n
2
∞ ( n ) Hn
4.8.7
. . . . . . . . . . . . . . . . . . . . .
n=1 4n n2
Harmonic Series with Binomial Coefficient in the Denominator
∞
4n Hn
4.9.1
. . . . . . . . . . . . . . . . . . . . .
n=1 (2n) n2
n
∞
4n H2n
4.9.2
n=1 (2n) n2 . . . . . . . . . . . . . . . . . . . . .
n
∞
4n Hn
4.9.3
. . . . . . . . . . . . . . . . . . . . .
2n
n=1 ( ) n3
n
. . . 290
. . . 291
. . . 291
. . . 292
. . . 292
. . . 293
. . . 294
. . . 295
. . . 297
. . . 299
. . . 299
. . . 301
. . . 302
Contents
xv
4.9.4
4.9.5
4.9.6
(2)
∞
4n Hn
n=1 (2n) n2
n
n
2
Hn
n2
∞
4
n=1 (2n)
n
∞
4n H2n
2n
n=1 ( ) n3
. . . . . . . . . . . . . . . . . . . . . . . . 304
. . . . . . . . . . . . . . . . . . . . . . . . 305
. . . . . . . . . . . . . . . . . . . . . . . . 307
n
Table of Mathematica Commands
312
References
314
Index
316
Notations
C
R
Z
Z≥0
Z≤0
Z+
Z−
R(z)
J(z)
n!
a
b
Γ
B
ζ
η
β
Lin
Hn
(a)
Hn
Hn
The set of complex numbers
The set of real numbers
The set of integers (Z = {. . . , −3, −2, −1, 0, 1, 2, 3, . . . })
The set of non-negative integers (Z≥0 = {0, 1, 2, . . . })
The set of non-positive integers (Z≤0 = {. . . , −2, −1, 0})
The set of positive integers (Z+ = {1, 2, 3, . . . })
The set of negative integers (Z− = {. . . , −3, −2, −1})
The real part of a complex number z
The imaginary part of a complex number z
n factorial
n
n! = 1 · 2 · 3 · · · (n − 1) · n = k=1 k, n ∈ Z+
The central binomial coefficient
Γ(a+1)
a
b = Γ(b+1)Γ(a−b+1)
The Gamma function
∞
Γ(z) = 0 tz−1 e−t dt, R(z) > 0
The Beta function
1
B(a, b) = 0 xa−1 (1 − x)b−1 dx, R(a) > 0, R(b) > 0
The Riemann zeta function
∞
ζ(z) = 1 + 21z + 31z + · · · = k=1 k1z , R(z) > 1
The Dirichlet eta function
k−1
∞
η(z) = 1 − 21z + 31z − · · · = k=1 (−1)
, R(z) > 0
kz
The Dirichlet beta function
∞
(−1)k
β(z) = 1 − 31z + 51z − · · · = k=0 (2k+1)
R(z) > 0
z,
The Polylogarithm function
2
3
k
∞
Lin (z) = z + 2zn + 3zn + · · · = k=1 kzn , |z| ≤ 1
The n-th harmonic number
n
Hn = 1 + 21 + 13 + · · · + n1 = k=1 k1 , n ∈ Z+
The n-th harmonic number of order a
(a)
n
Hn = 1 + 21a + 31a + · · · + n1a = k=1 k1a , n ∈ Z+
The skew harmonic number
n−1
k−1
n
H n = 1 − 12 + 13 − · · · + (−1)n
= k=1 (−1)k , n ∈ Z+
xvi
Notations
ψ
ψ (a)
ln
e
γ
G
xvii
The Digamma function
d
ψ(n) = dn
ln(Γ(n)) =
ln (Γ(n))
Γ(n)
The Polygamma function
da
da+1
ψ (a) (n) = dn
a ψ(n) = dna+1 ln(Γ(n))
The natural logarithm (loge )
The base of the natural logarithm
e = limn→∞ (1 + 1/n)n = 2.7182818284590 . . .
The Euler–Mascheroni constant
γ = limn→∞ (Hn − ln(n)) = 0.5772156649015 . . .
The Catalan’s constant
∞
(−1)k
G = 1 − 312 + 512 − · · · = k=0 (2k+1)
2 = 0.9159655941772 . . .
Chapter 1
Series Transformations and
Special Functions
1.1
Shifting the Sum Index
n+c
n
ak−c .
ak =
Proof. The index k in
n
k=m
(1.1)
k=m+c
k=m
ak ranges from m to n:
m ≤ k ≤ n.
Replace k by j − c,
m ≤ j − c ≤ n.
On solving this compound inequality, we get
m + c ≤ j ≤ n + c.
This indicates that if we replace the index k by j − c, the index j will range from
m + c to n + c:
n
n+c
ak =
k=m
aj−c .
j=m+c
Replace j by k in the latter equality to finish the proof.
1
2
Chapter 1. Series Transformations and Special Functions
Example 1: Let ak =
Hk
k+1
and m = 0 then shift the index by −1,
n
k=0
Hk
=
k+1
n+1
k=1
Hk−1
.
k
Example 2: Let ak = Hk xk−1 and m = 3 then shift the index by +2,
n−2
n
Hk+2 xk+1 .
Hk xk−1 =
k=1
k=3
1.2
Reversing the Order of the Sum Terms
n
n
an−k+m .
ak =
(1.2)
k=m
k=m
Proof. Following the previous proof, we have
m ≤ k ≤ n.
Replace k by n − j + m,
m≤n+m−j ≤n
or
m ≤ j ≤ n.
This shows that if we replace the index k by n − j + m, the index j will range from
m to n as well:
n
n
ak =
k=m
an−j+m .
j=m
The proof completes on replacing j by k in the latter equality.
This type of transformation reverses the order of the sum terms. To see that, let
m = 1 and n = 4 in (1.2), the LHS sum gives
4
ak = a1 + a2 + a3 + a4 ,
k=1
which is equivalent to the RHS sum:
4
a5−k = a4 + a3 + a2 + a1 ,
k=1
1.3. Splitting a Sum Into its Odd and Even Parts
3
but in reversed order.
Example 1: Put ak =
1
k
and m = 1,
n
k=1
1
=
k
n
k=1
1
.
n−k+1
(1.3)
Example 2: Put ak = k 2 and m = 3,
n
n
k2 =
k=3
1.3
(n − k + 3)2 .
k=3
Splitting a Sum Into its Odd and Even Parts
∞
∞
∞
(1.4)
n=1
n=0
n=1
a2n .
a2n+1 +
an =
Proof.
∞
an = a1 + a2 + a3 + · · ·
n=1
= (a1 + a3 + a5 + · · · ) + (a2 + a4 + a6 + · · · )
∞
∞
a2n ,
a2n+1 +
=
n=1
n=0
and the proof is complete
Example 1: Put an =
∞
1
n2 ,
∞
∞
∞
∞
1
1
1
1
1
1
=
+
=
+
2
2
2
2
2
n
(2n
+
1)
(2n)
(2n
+
1)
4
n
n=1
n=1
n=0
n=1
n=0
or
∞
∞
1
1
4
=
.
2
n
3 n=0 (2n + 1)2
n=1
Example 2: Put an =
Hn
n3 ,
∞
∞
∞
Hn
H2n+1
H2n
=
+
.
3
3
3
n
(2n
+
1)
(2n)
n=1
n=0
n=1
4
Chapter 1. Series Transformations and Special Functions
Converting the Summand a2n to an
1.4
∞
∞
a2n =
n=1
∞
1
1
an +
(−1)n an .
2 n=1
2 n=1
(1.5)
Proof. Starting with the RHS,
∞
∞
(−1)n an = a1 + a2 + a3 + · · · + (−a1 + a2 − a3 + · · · )
an +
n=1
n=1
= 2a2 + 2a4 + 2a6 + · · ·
∞
= 2(a2 + a4 + a6 + · · · ) = 2
a2n .
n=1
The proof finalizes on dividing both sides by 2. Following the same approach, we
also find
∞
∞
∞
1
1
a2n =
an +
(−1)n an .
(1.6)
2 n=0
2 n=0
n=0
Example 1: Let an =
1
(n+1)4
in (1.5),
∞
∞
∞
1
1
(−1)n
1
1
=
+
.
4
4
(2n + 1)
2 n=1 (n + 1)
2 n=1 (n + 1)4
n=1
Example 2: Let an =
Hn+1
(n+3)3
in (1.6),
∞
∞
∞
H2n+1
1
Hn+1
1
Hn+1
=
+
(−1)n
.
3
3
(2n + 3)
2 n=0 (n + 3)
2 n=0
(n + 3)3
n=0
Converting the Summand a2n+1 to an
1.5
∞
∞
a2n+1 =
n=0
∞
1
1
an −
(−1)n an .
2 n=1
2 n=1
(1.7)
Proof.
∞
∞
(−1)n an = a1 + a2 + a3 + · · · − (−a1 + a2 − a3 + · · · )
an −
n=1
n=1
= 2a1 + 2a3 + 2a5 + · · ·
1.6. Converting the Summand (−1)n a2n to in an
5
∞
= 2(a1 + a3 + a5 + · · · ) = 2
a2n+1 .
n=0
Divide both sides by 2 to complete the proof.
Let’s shift the index of the LHS sum in (1.8) by −1,
∞
∞
a2n−1 =
n=1
Example 1: Set an =
Hn
n3
∞
1
1
an −
(−1)n an .
2 n=1
2 n=1
(1.8)
in (1.7),
∞
∞
∞
Hn
H2n+1
1
1
Hn
(−1)n 3 .
=
−
3
3
(2n + 1)
2 n=1 n
2 n=1
n
n=0
Example 2: Set an =
1
n4
in (1.8),
∞
∞
∞
1
(−1)n
1
1
1
=
−
.
4
4
(2n − 1)
2 n=1 n
2 n=1 n4
n=1
1.6
Converting the Summand (−1)n a2n to in an
∞
∞
in an .
n
(−1) a2n = R
(1.9)
n=1
n=1
Proof.
∞
in an = ia1 + i2 a2 + i3 a3 + i4 a4 + i5 a5 + i6 a6 + · · ·
n=1
= ia1 − a2 − ia3 + a4 + ia5 − a6 + · · ·
= i(a1 − a3 + a5 − · · · ) + (−a2 + a4 − a6 + · · · )
∞
∞
(−1)n a2n+1 +
=i
n=0
(−1)n a2n ,
n=1
and the proof follows on comparing the real parts of both sides.
Example 1: Put an =
xn
n3 ,
∞
∞
(−1)n
n=1
n
x2n
nx
=
R
i
.
(2n)3
n3
n=1
(1.10)
6
Chapter 1. Series Transformations and Special Functions
Example 2: Put an =
Hn+1
n2 ,
∞
H2n+1
(−1)n
(2n)2
n=1
1.7
∞
in
=R
n=1
Hn+1
.
n2
Converting the Summand (−1)n a2n+1 to in an
∞
∞
in an .
(−1)n a2n+1 = J
(1.11)
n=1
n=0
Proof. Compare the imaginary parts of both sides of (1.10).
Example 1: Let an =
1
n3 ,
∞
∞
(−1)n
in
=
J
.
(2n + 1)3
n3
n=0
n=1
Example 2: Let an =
Hn
(n+1)2 ,
∞
∞
(−1)n
n=0
1.8
H2n+1
Hn
in
=J
.
(2n + 2)2
(n
+ 1)2
n=1
Converting a Sum to a Product
r
r
ln(an ) = ln
n=m
an .
(1.12)
n=m
Proof.
r
ln(an ) = ln(am ) + ln(am+1 ) + · · · + ln(ar )
n=m
r
= ln(am × am+1 × · · · × ar ) = ln
an .
n=m
Example 1: Let an = n,
r
r
n = ln(1 × 2 × 3 × · · · × r) = ln(r!).
ln(n) = ln
n=1
n=1
1.9. Double Summation
7
Example 2: Let an = en ,
r
r
n=1
1.9
r
en =
ln
ln(en ) =
n=
n=1
n=1
r(r + 1)
.
2
Double Summation
∞
∞
m
∞
a m bn =
am bn .
(1.13)
n=1 m=n
m=1 n=1
Proof.
∞
m
1
am bn = a1
m=1 n=1
2
3
bn + a2
n=1
bn + · · ·
bn + a3
n=1
n=1
= a1 (b1 ) + a2 (b1 + b2 ) + a3 (b1 + b2 + b3 ) + · · ·
= b1 (a1 + a2 + · · · ) + b2 (a2 + a3 + · · · ) + b3 (a3 + a4 + · · · ) + · · ·
∞
= b1
∞
∞
am + b2
m=1
∞
m=2
∞
=
am + · · ·
am + b3
bn
∞
m=3
∞
am bn ,
am =
n=1 m=n
m=n
n=1
and the proof is complete. If we follow the same steps above, we also find
∞ m−1
∞
∞
am bn =
m=1 n=1
am bn .
n=1 m=n+1
Example 1: Let am = pm and bn = pn ,
∞
∞
m
pm+n =
m=1 n=1
∞
pn
n=1
pm
m=n
{use the geometric series formula for the inner sum asuming |p| < 1}
∞
pn
=
n=1
pn
1−p
∞
=
1
(p2 )n
1 − p n=1
{use the geometric series formula again}
=
1
1−p
p2
1 − p2
=
p2
.
(1 − p)(1 − p2 )
(1.14)
