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CHDISMTHAPP
 Methods for the
Summation of Series

Tian-Xiao He
MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks
does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of
MATLAB® software or related products does not constitute endorsement or sponsorship by The
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ISBN: 978-0-367-50797-8 (hbk)

ISBN: 978-1-032-19500-1 (pbk)
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DOI: 10.1201/9781003051305

Publisher’s note: This book has been prepared from camera-ready copy provided by the authors
Contents

Foreword xi

Testimonial xiii

Preface xv

Biography xix

Symbols xxi

1 Classical Methods from Infinitesimal Calculus 1

1.1 Use of Infinitesimal Calculus . . . . . . . . . . . . . . . . . . 2
1.1.1 Convergence of series . . . . . . . . . . . . . . . . . . . 2
1.1.2 Limits of sequences and series . . . . . . . . . . . . . . 8
1.2 Abel’s Summation . . . . . . . . . . . . . . . . . . . . . . . . 20
1.2.1 Abel’s theorem and Tauber theorem . . . . . . . . . . 20
1.2.2 Abel’s summation method . . . . . . . . . . . . . . . . 27
1.3 Series Method . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1.3.1 Use of the calculus of finite difference . . . . . . . . . 36
1.3.2 Application of Euler-Maclaurin formula and the
Bernoulli polynomials . . . . . . . . . . . . . . . . . . 48

2 Symbolic Methods 67
2.1 Symbolic Approach to Summation Formulas of Power Series 68
2.1.1 Wellknown symbolic expressions . . . . . . . . . . . . 69
2.1.2 Summation formulas related to the operator
(1 − xE)−1 . . . . . . . . . . . . . . . . . . . . . . . . 74
2.1.3 Consequences and examples . . . . . . . . . . . . . . . 77
2.1.4 Remainders of summation formulas . . . . . . . . . . . 82
2.1.5 Q-analog of symbolic operators . . . . . . . . . . . . . 86
2.2 Series Transformation . . . . . . . . . . . . . . . . . . . . . . 96
2.2.1 An extension of Eulerian fractions . . . . . . . . . . . 98
2.2.2 Series-transformation formulas . . . . . . . . . . . . . 99
2.2.3 Illustrative examples . . . . . . . . . . . . . . . . . . . 105
2.3 Summation of Operators . . . . . . . . . . . . . . . . . . . . 112

v
vi Contents

2.3.1 Summation formulas involving operators . . . . . . . . 112

2.3.2 Some special convolved polynomial sums . . . . . . . . 120
2.3.3 Convolution of polynomials and two types of
summations . . . . . . . . . . . . . . . . . . . . . . . . 123
2.3.4 Multifold Convolutions . . . . . . . . . . . . . . . . . . 125
2.3.5 Some operator summation formulas from multifold
convolutions . . . . . . . . . . . . . . . . . . . . . . . . 130

3 Source Formulas for Symbolic Methods 137

3.1 An Application of Mullin-Rota’s Theory of Binomial
Enumeration . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
3.1.1 A substitution rule and its scope of applications . . . 138
3.1.2 Various symbolic operational formulas . . . . . . . . . 141
3.1.3 Some theorems on Convergence . . . . . . . . . . . . . 143
3.1.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 150
3.2 On a Pair of Operator Series Expansions Implying a Variety of
Summation Formulas . . . . . . . . . . . . . . . . . . . . . . 156
3.2.1 A pair of (∞4 ) degree formulas . . . . . . . . . . . . . 157
3.2.2 Specializations and examples . . . . . . . . . . . . . . 160
3.2.3 A further investigation of a source formula . . . . . . 168
3.2.4 Various consequences of the source formula . . . . . . 171
3.2.5 Lifting process and formula chains . . . . . . . . . . . 177
3.3 (Σ∆D) General Source Formula and Its Applications . . . . 178
3.3.1 (Σ∆D) GSF . . . . . . . . . . . . . . . . . . . . . . . 178
3.3.2 GSF implies SF(2) and SF(3) . . . . . . . . . . . . . . 182
3.3.3 Embedding techniques and remarks . . . . . . . . . . 183

4 Methods of Using Special Function Sequences, Number

Sequences, and Riordan Arrays 193
4.1 Use of Stirling Numbers, Generalized Stirling Numbers, and
Eulerian Numbers . . . . . . . . . . . . . . . . . . . . . . . . 194
4.1.1 Basic convergence theorem . . . . . . . . . . . . . . . 194
4.1.2 Summation formulas involving Stirling numbers,
Bernoulli numbers, and Fibonacci numbers . . . . . . 200
4.1.3 Summation formulas involving generalized Eulerian
functions . . . . . . . . . . . . . . . . . . . . . . . . . 205
4.2 Summation of Series Involving Other Famous Number
Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
4.2.1 Convergence theorem and examples . . . . . . . . . . 210
4.2.2 More summation formulas involving Fibonacci numbers
and generalized Stirling numbers . . . . . . . . . . . . 216
4.2.3 Summation formulas of (ΣS) class . . . . . . . . . . . 225
4.3 Summation Formulas Related to Riordan Arrays . . . . . . . 235
Contents vii

4.3.1 Riordan arrays, the Riordan group, and their sequence

characterizations . . . . . . . . . . . . . . . . . . . . . 236
4.3.2 Identities generated by using extended Riordan arrays
and Faà di Bruno’s formula . . . . . . . . . . . . . . . 247
4.3.3 Various row sums of Riordan arrays . . . . . . . . . . 263
4.3.4 Identities generated by using improper or non-regular
Riordan arrays . . . . . . . . . . . . . . . . . . . . . . 272
4.3.5 Identities related to recursive sequences of order 2 and
Girard-Waring identities . . . . . . . . . . . . . . . . . 279
4.3.6 Summation formulas related to Fuss-Catalan numbers 285
4.3.7 One-pth Riordan arrays and Andrews’ identities . . . 295

5 Extension Methods 305

5.1 Identities and Inverse Relations Related to Generalized Riordan
Arrays and Sheffer Polynomial Sequences . . . . . . . . . . . 306
5.1.1 Generalized Riordan arrays and the recurrence relations
of their entries . . . . . . . . . . . . . . . . . . . . . . 308
5.1.2 The Sheffer group and the Riordan group . . . . . . . 318
5.1.3 Higher dimensional extension of the Riordan group . . 329
5.1.4 Dual sequences and pseudo-Riordan involutions . . . . 342
5.2 On an Extension of Riordan Array and Its Application in the
Construction of Convolution-type and Abel-type Identities . 354
5.2.1 Generalized Riordan arrays with respect to basic
sequences of polynomials . . . . . . . . . . . . . . . . 354
5.2.2 A general class of convolution-type identities . . . . . 364
5.2.3 A general class of Abel identities . . . . . . . . . . . . 374
5.3 Various Methods for constructing Identities Related to
Bernoulli and Euler Polynomials . . . . . . . . . . . . . . . . 384
5.3.1 Applications of dual sequences to Bernoulli and Euler
polynomials . . . . . . . . . . . . . . . . . . . . . . . . 387
5.3.2 Extended Zeilberger’s algorithm for constructing
identities related to Bernoulli and Euler polynomials . 399
5.3.2.1 Gosper’s algorithm . . . . . . . . . . . . . . . 399
5.3.2.2 W-Z algorithm . . . . . . . . . . . . . . . . . 402
5.3.2.3 Zeilberger’s creative telescoping algorithm . . 403
5.3.2.4 Extended Zeilberger’s algorithm . . . . . . . 405

Bibliography 417

Index 433
 To Yulan
To Calvin and Viola
Foreword

You are about to encounter a very special book. Summing series has been of
interest for centuries, and, in an age of powerful computers, the interest has
greatly intensified. Keopf’s Hypergeometric Summations, The Concrete Tetra-
hedron by Kauers and Paule, and A=B by Petkovsek, Wilf, and Zeilberger
are all impressive works devoted to this topic.
So why do we need, Methods for the Summation of Series, ostensibly de-
voted to the same subject? Let us begin by noting the background that the
author Tian-Xiao He (Earl and Marian A. Beling Professor of Natural Sci-
ences and Professor of Mathematics, Illinois Wesleyan University) brings to
this effort. He has done important work in numerical analysis, wavelet analy-
sis, approximation theory, and splines. These interests have led him naturally
into enumerative combinatorics and the emerging field of Riordan Arrays. This
diversity of interests is on full display in this book. It would be fair to say that
this volume combines the charm of an ancient book like I. J. Schwatt’s, An
Introduction to the Operations with Series (1924), with a keen awareness of
the many aspects of the most recent methods developed for the summation of
series. The advantage of this mixture is that insight and context are provided
for many applications.
The five chapters of this book provide a clear view of the depth of vision.
The first chapter is devoted to classical methods, which, while they date back
to the 19th century and before, are nonetheless effective and always timely.
Symbolic methods occupy the next two chapters. This, too, is a venerable
subject dating back to invariant theory; its modern combinatorial manifesta-
tions were pioneered by Gian-Carlo Rota. This is a compelling way to place
the classic theory of finite differences in a modern and substantially more
powerful setting.
Chapter 4 moves to the world of special functions. Of particular interest
is the extensive use of Riordan Arrays, a topic in which Professor He is one of
the world leaders. This is, indeed, one of the highlights of this book. The final
chapter continues to build on Riordan Arrays and concludes with an account
of some of the algorithms that have been so successful in doing summations
via computer algebra.

xi
xii Foreword

This is a well-written, lucid book with many surprising gems. I am happy

to recommend it to you as a valuable addition to your library.
George E. Andrews
Evan Pugh University Professor in Mathematics
Member, National Academy of Sciences (USA)
Past President, American Mathematical Society
Testimonial

In the past three months, I really enjoyed reading through the book. It is a
very good monograph and text and offering an overview of several valuable
techniques, and readers will find it to be a very fine reference book as well as
one from which to study. I certainly give it my highest recommendation. The
author presented very impressive publications and research activities.
Henry Wadsworth Gould
Professor Emeritus of Mathematics
West Virginia University, Morgantown
Fellow of the American Association for the Advancement of Science
Honorary Fellow of the Institute of Combinatorics and its Applications
July 11, 2021

xiii
Preface

This book presents methods for the summation of infinite and finite series and
the related identities and inversion relations. The summation includes the col-
umn sums and row sums of lower triangular matrices. The convergence of the
summation of infinite series is considered. We focus on symbolic methods and
the Riordan array approach for the summation. Much of the materials in this
book have never appeared before in textbook form. This book can be used as
a suitable textbook for advanced courses for higher-level undergraduate and
lower level graduate students. It is also an introductory self-study book for
researchers interested in this field, while some materials of the book can be
used as a portal for further research. In addition, this book contains hundreds
of summation formulas and identities, which can be used as a handbook for
people working in computer science, applied mathematics, and computational
mathematics, particularly, combinatorics, computational discrete mathemat-
ics, and computational number theory. The exercises at the end of each chapter
help deepen understanding.
Since the methods discussed in this book are related to the classical sum-
mation methods, we present the main classical methods in Chapter 1 with the
example oriented way. This chapter provides useful supplementary materials
for the people who study advanced Calculus, and training materials for the
people who study applied and computational mathematics.
The infinitesimal calculus or differential and integral calculus is a field to
treat functions of continuous independent variables. The methods to find sum-
mation of series by using infinitesimal calculus shall be surveyed Chapter 1.
We will introduce one by one the following five simple and common methods:
(1) Substitution method; (2) Telescoping method; (3) Method of the sum-
mation of trigonometric series; (4) Differentiation and integration method for
uniformly convergent series; and (5) Abel’s summation.
As is well known, the closed form representation of series has been studied
extensively. It is also known that the symbolic calculus with operators ∆ (dif-
ference), E (shift or displacement), and D (derivative) plays an important role
in the Calculus of Finite Differences, which is often employed by statisticians
and numerical analysts. The object of Chapter 2 is to make use of the classical
operators ∆, E, and D to develop closed forms for the summation of power se-
ries that appear to have a certain wide scope of applications. Throughout this
chapter the theory of formal power series and of differential operators will be
utilized, while the convergence of the infinite series is discussed. In this chap-
ter, we focus on the summation and identities arising from the interrelations

xv
xvi Preface

of a number of operators in common use in combinatorics, number theory, and

discrete mathematics. Various well-known results can also be found in some
classical treatises in this chapter. Since all the symbolic expressions used and
operated in the calculus could be formally expressed as power series in ∆ (or
D or E) over the real or complex number field, it is clear that the theoretical
basis of the calculus may be found within the general theory of the formal
power series. Worth reading is a sketch of the theory of formal series that has
been given briefly in Chapter 2.
Chapter 3 presents a frame work with several source formulas, from which
numerous summation formulas and identities are constructed. This frame work
is due to a general substitution rule, called Mullin-Rota’s substitution rule.
Given a generating function or a formal power series expansion, then a cer-
tain operational formula may be obtained by using the substitution rule. Some
operator summation formulas from multifold convolutions are also obtained
similarly. With the aid of Mullin-Rota’s substitution rule, we shall show in this
chapter that the Sheffer-type differential operators together with the delta op-
erators ∆ and D could be used to construct a pair of expansion formulas that
imply a wide variety of summation formulas in the discrete analysis and com-
binatorics. A convergence theorem is established for fruitful source formulas.
Numerous new formulas are represented as illustrative examples. A kind of
lifting process is used to enlarge the number of new formulas. In addition, this
chapter presents further investigation on a general source formula (GSF) that
has been proved capable of deducing numerous classical and new formulas for
series expansions and summations besides those given in the previous parts of
the book.
In the first half of Chapter 4, we shall continue the symbolic process for
some special function sequences and number sequences such as the sequences
of Bernoulli polynomials and numbers, Stirling numbers, Fibonacci numbers,
etc. In the second half of this chapter, we construct identities and summation
formulas for the function sequences and number sequences related to Riordan
arrays. A Riordan array is an infinite lower triangular matrix, which columns
are multiplication of certain power series g and f . The theory of Riordan
arrays provides a modern method for classical umbra calculus, bringing new
insights into many areas of combinatorial importance. This chapter gives an
introduction to the basic and applicable materials on Riordan arrays and the
Riordan group for students and researchers, who seek novel ways of working in
fields such as combinatorial identities, triangles for enumerating combinatorial
numbers, special polynomial sequences, orthogonal polynomials, etc.
In Chapter 5, we shall present the methods extended from the previous
chapters for constructing various summation formulas, identities, and inver-
sion relations. The formulas constructed in this chapter include the identi-
ties of high dimensions, convolution-type, Abel-type, and those related to
Bernoulli polynomials and numbers, Euler polynomials and numbers, etc.
Some methods represented in this chapter are related to generalized Riordan
arrays and generalized Riordan groups with different bases and their Sheffer
Preface xvii

analogs, Sheffer-type polynomial sequences and the Sheffer group. Some iden-
tities and inversion relations are constructed by using dual sequences with the
Riordan array representation named pseudo-Riordan involutions. Finally, an
extension of W-Z algorithm and Zeilberger’s creative telescoping algorithm
is represented and used to construct and prove the identities for Bernoulli
polynomials and numbers.
I am grateful to Professor Leetsch C. Hsu (Xu Lizhi) for guiding me into
the field of enumeration combinatorics. This book is dedicated to the mem-
ory of him. The author would like to thank Professor George Andrews for
his foreword and Professor Henry Gould for his testimonial and both of them
for their comments and encouragements. I would like to thank all the col-
laborators who have published joint papers with me in the fields of symbolic
methods and/or Riordan arrays over the past decades, especially Leetsch C.
Hsu, Louis W. Shapiro, Renzo Sprugnoli, Henry W. Gould, and Peter J.-S.
Shiue for the pleasant cooperation with them and everything I have learned
from them. I would like to thank the Editors of the Discrete Mathematics and
Its Applications Series, CRC Press/Taylor & Francis Group, LLC, and KGL,
specially Miklos Bona, Robert Ross, Vaishali Singh, and Manisha Singh for
their help and patience in the process. The author is thankful for the support
given by the Earl and Marian A. Beling Professor’s Fund.
Tian-Xiao He
Illinois Wesleyan University
Bloomington, IL
Biography

Tian-Xiao He received Ph.D. degrees at the Dalian University of Technology,

Dalian, and Texas A&M University, College Station, respectively. Dr. He is a
Professor of Mathematics and Earl and Marian A. Beling Professor of Natural
Science at Illinois Wesleyan University. Dr. He has authored or co-authored
over 150 research articles and 7 volumes in mathematics and is presently an
editor/chief editor for several math journals.

xix
Symbols

Symbol Description

N Natural number set (g, f ) The Riordan array gener-

N0 The set of natural num- ated by g(t) ∈ F 0 , f (t) ∈
bers and zero F1
n
Z The ring of integers Unsigned Stirling num-
k
R The field of real numbers bers of the first kind
C The field of complex num- n
k Stirling numbers of the
bers second kind
F = K[[t]] The ring of formal power
series in variable t over a S(n, k) Generalized Stirling num-
field K. bers
Fr The set of formal power C(t) The generating function
series of order r of the Catalan numbers
∆ Difference Fm (t) The generating function
E Displacement of the Fuss-Catalan num-
D Derivative bers

xxi
1
Classical Methods from Infinitesimal
Calculus

CONTENTS
1.1 Use of Infinitesimal Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Convergence of series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 Limits of sequences and series . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2 Abel’s Summation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.2.1 Abel’s theorem and Tauber theorem . . . . . . . . . . . . . . . . . . . . 20
1.2.2 Abel’s summation method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.3 Series Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1.3.1 Use of the calculus of finite difference . . . . . . . . . . . . . . . . . . . 36
1.3.2 Application of Euler-Maclaurin formula and the
Bernoulli polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Since the methods discussed in this book are related to the classical summation
methods, we start this book from main classical methods for summation of
series described in an example-oriented way.
The infinitesimal calculus or differential and integral calculus is a field to
treat functions of continuous independent variables, i.e., the variables may take
every possible value in a given interval. The methods to find the summation
of series by using infinitesimal calculus shall be surveyed in this chapter in the
example-oriented way. P
The summation of a series n≥1 an is defined by

X n
X
an = lim sn ≡ lim ak ,
n→∞ n→∞
n≥1 k=1
Pn
where sn = k=1 ak is the partial sum of the series. On the summation of
infinite series, there are following five simple and common methods.
P
(1) Substitution method. To get the sum ofP n≥0 un , we may substitute
un = anP k n that brings the sum f (k) = n≥0 un if a known function
f (x) = n≥0 an xn can be determined.
P
(2) Telescoping method. In a given series n≥0 un , if we have Pun = vn −
vn+1 (n = 0, 1, . . .) and limn→∞ vn = v∞ < ∞, then n≥0 un =

DOI: 10.1201/9781003051305-1 1
2 Methods for the Summation of Series

v0 − v∞ . In particular, if
1
un = ,
an an+1 · · · an+m
where ak = c + kd (k = 1, 2, . . .), c, d ∈ R, d 6= 0, then
1 1
vn = .
md an an+1 · · · an+m−1
(3) Trigonometric series summation. In order to evaluate the sums of
X X
an cos(nx) and an sin(nx),
n≥0 n≥0

we consider them as the real part and the imaginary part of power
series X
an z n ,
n≥0
−ix
where z = e , which can be summarized. In many cases, summa-
tion X1 1
z n = log (|z| < 1)
n 1−z
n≥0

is useful to find the sum of the power series.

(4) Differentiation and integration method for uniformly convergent se-
ries. Namely, we may transfer a given uniform convergent series to
a new series by differentiation or a suitable integration that can be
summed, and the sum of original series will be found by taking an
inverse transformation, i.e., integration or differentiation.
(5) Abel’s summation. One may see Subsection 1.2.2 for details.
Some examples shown in this chapter are selected from or greatly influ-
enced by [1, 44, 51, 73, 79, 84, 102, 123, 135, 139, 141, 143, 164, 165, 177, 183,
191, 200].

1.1 Use of Infinitesimal Calculus

1.1.1 Convergence of series
Example 1.1.1 Here is an example of the first method. Since
X (−1)n−1
= ln 2, (1.1)
n
n≥1

we immediately have
X (−1)n−1 X   X (−1)n−1
n−1 1 1
= (−1) − =2 − 1 = 2 ln 2 − 1.
n(n + 1) n n+1 n
n≥1 n≥1 n≥1
Classical Methods from Infinitesimal Calculus 3

A few examples for the second method dealing with the so-called telescop-
ing series are shown below.
Example 1.1.2 Since
 
1 1 1 1
= − ,
n(n + 1)(n + 2) 2 n(n + 1) (n + 1)(n + 2)

we obtain X 1 1
= .
n(n + 1)(n + 2) 4
n≥1

For m ∈ N, we have
 
1 1 1 1
= − .
n(n + m) m n n+m

Thus,  
X 1 1 1 1
= 1 + + ...+ .
n(n + m) m 2 m
n≥1

Similarly,

X Xℓ  
2n + 1 1 1
= lim − = 1.
n2 (n + 1)2 ℓ→∞
n=1
n2 (n + 1)2
n≥1

Telescoping method can also be used for the following triangular function
series.  
X 1 π
tan−1 = .
n2 + n + 1 4
n≥1

In fact,

X   ℓ
X
−1 1
tan 2
= lim [tan−1 (n + 1) − tan−1 n]
n +n+1 ℓ→∞
n=1
n≥1
−1 −1 π
= lim [tan (ℓ + 1) − tan 1] = .
ℓ→∞ 4
Similarly,  
X 2n + 1 π
−1
tan = .
n (n + 1)2
2 4
n≥1

Some series can be transferred to telescoping series as shown below.

4 Methods for the Summation of Series

Example 1.1.3
X 1 X1 1
2
= −
n2 (n + 1)2 n n+1
n≥1 n≥1
X 1 2 1

= − +
n2 n(n + 1) (n + 1)2
n≥1

π2
= − 3,
3
P P
where we use n≥1 1/n2 = π 2 /6, and the telescoping series n≥1 1/n (n +
1) = 1.
Similarly,
X n
(n + 1)(n + 2)(n + 3)
n≥1
X 1 X 1
= −
(n + 2)(n + 3) (n + 1)(n + 2)(n + 3)
n≥1 n≥1
1 1 1
= − = .
3 12 4
Example 1.1.4 Using the method of differentiation term by term to the fol-
lowing series in their uniform convergence intervals, |x| < 1, we obtain
 ′′  ′
X X X
n2 xn−1 =  xn+1  −  xn 
n≥1 n≥1 n≥1
 ′′  ′
1 1 1+x
= −1−x − −1 = .
1−x 1−x (1 − x)3

Similarly,
X x(3 − x)
n(n + 2)xn = (|x| < 1) and
(1 − x)3
n≥1
X 2n + 1 2
x2n = (1 + 2x2 )ex ,
n!
n≥1

where the Taylor’s series of 1/(1 − x) and ex are applied.

Some techniques shown below are useful in applying differentiation

method. Consider the summation of the following series
X a(a + d) . . . [a + (n − 1)d]
y(x) := xn ,
d(2d) · · · (nd)
n≥1
Classical Methods from Infinitesimal Calculus 5

d > 0. It is easy to have the differential equation

a
(1 − x)y ′ (x) = y(x),
d
which yields a solution y(x) = (1 − x)−a/d .
Another example for differentiation method is

1 X ((n − 1)!)2
(2x)2n = (sin−1 x)2
2 (2n)!
n≥1

for all |x| < 1. Denote y ≡ y(x) = (sin−1 x)2 . it is easy to find y satisfies
differential equation (1 − x2 )y ′′ − xy ′ − 2 = 0 and one of its power series
solution is
1 X ((n − 1)!)2
y= (2x)2n .
2 (2n)!
n≥1

Readers can check the correction by evaluating the first few terms of Taylor’s
expansion of (sin−1 (x))2 and compare with the series shown above. Another
power series solution of the differential equation can be obtained using coeffi-
cient comparison method.
Making use of the complex function properties, one may obtain more
summation formulas. Denote by w = u + iv a complex number. Then
log w = log |w| + i(arg w + kπ), where arg w = tan−1 (v/u) is an argument
of w, and k is an arbitrary integer. Let Re(w) be the real part of w. Then
Re(log w) = log |w|. Denote z = eix . We have
 
X cos nx X zn
= Re   = −Re(log(1 − z))
n n
n≥1 n≥1

= −Re(log(1 − eix )) = − log |1 − eix |.

Using Euler formula, we obtain

  
ix ix/2 1 −ix/2 ix/2 x
1 − e = 2e e −e = −2eix/2 sin .
2 2

Thus, X cos nx x
= − log 2 sin (1.2)
n 2
n≥1

for 0 < x < 2π. More examples are given as follows.

Example 1.1.5
 
X sin nx X zn
= Im   = π − x , (0 < x < 2π)
n n 2
n≥1 n≥1
6 Methods for the Summation of Series
 
X cos nx X zn
= Re   = Re(ez ) = ecos x cos(sin x), (|x| < ∞)
n! n!
n≥1 n≥1
 
X sin nx X zn
= Im   = Im(ez ) = ecos x sin(sin x) (|x| < ∞).
n! n!
n≥1 n≥1

Using formula sin na sin nx = 12 [cos n(a − x) − cos n(a + x)] and summation
(1.2) yields
X sin na sin nx 1 sin 12 (x + a)
= log .
n≥1
n 2 sin 12 (x − a)
Similarly, formula
1
sin2 na sin nx = (sin n(2a − x) − sin n(2a + x) + 2 sin nx)
4
and the first summation in Example 1.1.5 yields
X sin2 na sin nx π
= (1.3)
n 4
n≥1

for all 0 < x < 2a < π. By taking limit a → π/2 on the both sides of (1.3),
we obtain
X sin(2n − 1)x π
= sgnx, (|x| ≤ π). (1.4)
2n − 1 4
n≥1

From (1.4) one can establish

X cos(2n − 1)x π2 π
2
= − |x|, (|x| ≤ π). (1.5)
(2n − 1) 8 4
n≥1
P
Indeed, for |x| = π, (1.5) is easy to be obtained from n≥1 1/n2 = π 2 /6. For
|x| < π, denoting the series on the left-hand side of (1.5) by F (x) and taking
its derivative term by term yields
d X sin(2n − 1)x π
F (x) = − = − sgnx,
dx 2n − 1 4
n≥1

where the last step is from (1.4). The antiderivative of the above equation
generates Z
π π
F (x) = − sgnxdx = C − |x|,
4 4
where the constant C is determined by
X 1 π2
C = F (0) = = ,
(2n − 1)2 8
n≥1

which implies (1.5).

Classical Methods from Infinitesimal Calculus 7
P
We now consider a positive divergent series n≥1 1/an . Then, for x > 0,
we have
     
a1 a1 a2 a1 a2
+ +
a2 + x a2 + x a3 + x a2 + x a3 + x
 
a3 a1
× + ··· = . (1.6)
a4 + x x

Indeed, it is easy to write

     
a1 a1 a2 + x a1 a1 a2
= = +
x a2 + x x a2 + x a2 + x x
a1
= + R1 ,
a2 + x
      
a1 a1 a1 a2 a1 a2 a3
= + +
x a2 + x a2 + x a3 + x a2 + x a3 + x x
  
a1 a1 a2
= + + R2 .
a2 + x a2 + x a3 + x

In general, we have the expression of the remainder Rn as

    
a1 a2 a3 an+1
Rn = ···
x a2 + x a3 + x an+1 + x
 
a1 n+1 x
= Π 1− .
x k=2 ak + x

Hence, to prove the convergence of series of (1.6), we only need to show the
infinite product tends to zero as n → ∞.
Since x > 0, we have
X x X 1
= ,
ak + x bk + 1
k≥2 k≥2
P
where bk = ak /x (k = 2, 3, . . .). The divergence of k≥1 1/ak implies the
P
divergence of k≥1 1/bk . If lim bk < ∞, then limk→∞ 1/(bk + 1) 6= 0, which
P k→∞
implies that k≥2 1/(bk + 1) diverges to infinite. If lim bk = ∞, then 1/(bk +
P k→∞
1) ∼ 1/bk as k → ∞. Hence, k≥1 1/(bk + 1) = ∞. We have shown in any
case,
n+1  
a1 Y x
Rn = 1− →0
x ak + x
k=2

as n → ∞. This completes the proof of formula (1.6).

Formula (1.6) has a lot of applications. Here are two examples.
8 Methods for the Summation of Series

Example 1.1.6
1! 2! 3!
+ + ···
x + 1 (x + 1)(x + 2) (x + 1)(x + 2)(x + 3)
1
= , (x > 1)
x−1
x x2 x4
+ + + ···
1 − x2 1 − x4 1 − x8
(
x/(1 − x), if |x| < 1
=
1/(1 − x), if |x| > 1.

Here, the first formula is from (1.6) by substituting transform x → x − 1 and

an = n. And we leave them as exercises (cf. Exercise 1.4).

1.1.2 Limits of sequences and series

In this section, we discuss the limits of sequences and series, which will be ap-
plied in convergence of formal series and sequence approximation. The major
part of this section is Toeplitz theorem on sequence transformation and its
corollaries as well as the limits of sequences and series related to integrals.
First, we establish the following proposition.
Pn 1 Pn
Proposition
P 1.1.7 Denote sn = k=1 vk , tn = n k=1 sk , and τn =
1 n
n k=1 kv k . Then
(i) sn → ℓ implies tn → ℓ and τn → 0.
(ii) tn → ℓ and τn → τ imply τ = 0 and sn → ℓ.

Proof. (i) Write sn = ℓ + un . Thus for and ǫ > 0, there exists N ≡ N (ǫ) such
that n ≥ N implies |un | < ǫ/3. Thus, for n ≥ N , we have
N n N
1X 1 X 1X N
tn = sk + (ℓ + vk ) = sk + ℓ − ℓ + rn ,
n n n n
k=1 k=N +1 k=1

where
n
1 X n−N ǫ ǫ
|rn | ≡ uk ≤ · ≤ .
n n 3 3
k=N +1

For the fixed N chosen as above, there exits n0 ≥ N such that

N
1X ǫ Nℓ ǫ
sk < , < .
n 3 n 3
k=1

Therefore,
1 1 1
|tn − ℓ| < ǫ+ ǫ+ ǫ=ǫ
3 3 3
when n ≥ n0 , i.e., tn → ℓ as n → ∞.
Classical Methods from Infinitesimal Calculus 9

Since
1
τn = [−(s1 + s2 + · · · sn−1 ) + nsn ]
n
n−1
= − tn−1 + sn → −ℓ + ℓ = 0
n
when n → ∞.
(ii) We have shown that
1
tn + τn = (sn + nsn ),
n
or equivalently,
n
(tn + τn ).
sn =
n+1
Hence, from (i), τn → 0 and lim sn = lim tn = ℓ.
n→∞ n→∞

Proposition 1.1.8 If An → A and Bn → B as n → ∞, then

n
1X
Ak Bn+1−k → AB, (n → ∞).
n
k=1

Proof. For any ǫ > 0 (ǫ < 1), there exists N such that An = A + Rn and
Bn = B + Sn with |Rn |, |Sn | < ǫ whenever n ≥ N . Hence, for all p, q ≥ N , we
have
Ap Bq = AB + ASq + BRp + Rp Sq ,
|Ap Bq − AB| < ǫ(|A| + |B| + 1).
Let n ≥ 2N . Then
n
1X
Ak Bn−k+1
n
k=1
N
X −1 n−N +1 n
1 1 X 1 X
= Ak Bn−k+1 + Ak Bn−k+1 + Ak Bn−k+1 .
n n n
k=1 k=N k=n−N +2

Since Am , Bn are bounded, the first and the third sums on the right-hand
side of above equation tend to zero as n → ∞. In addition,
n−N +1
1 X 2(N − 1)
Ak Bn−k+1 = AB − AB + H,
n n
k=N

where |H| < ǫ(|A| + |B| + 1). Therefore, there exists n0 such that n ≥ n0 (ǫ)
implies
n
1X
Ak Bn−k+1 − AB < ǫ[1 + 1 + (|A| + |B| + 1) + 1],
n
k=1

which completes the proof of the proposition.

10 Methods for the Summation of Series

We now present the Toeplitz’s sequence transformation theorem. Consider

lower triangular matrix
p00
p10 p11
(1.7)
p20 p21 p22
··· ··· ··· ···
Pn
where pnk ≥ 0 and the sums of all rows are one (i.e., k=0 pnk = 1). For a
given sequence (sn ), we call sequence
n
X
tn = pnk sk , (1.8)
k=0

n = 0, 1, 2, . . ., a transformed sequence from (sn ) with respect to matrix

[pnk ]n≥k≥0 .
Theorem 1.1.9 (Toeplitz’s sequence transformation theorem) Assume (tn )
is transformed from (sn ) with respect to [pnk ]n≥k≥0 . Then
lim sn = s → lim tn = s
n→∞ n→∞

holds if and only if for every fixed m

lim pnm = 0.
n→∞

Proof. Necessity. If sn = 0 for n 6= m and sm = 1. Then tn = pnm (n ≥ m),

which yields
lim pnm = lim tn = lim sn = 0.
n→∞ n→∞ n→∞
Sufficiency. If the antecedent of theorem holds, then for any ǫ > 0 there
exists N ≡ N (ǫ) such that n > N implies |sn − s| < ǫ/2. In addition, there
exists N ′ > N so that
ǫ
pn0 , pn1 , . . . , pnN < ,
4(N + 1)M
are fulfilled for every n > N ′ , where M = M ax|sk | (note that if M = 0, then
the conclusion of the theorem is trivial). Therefore,
n
X
tn − s = pnk (sk − s)
k=0

satisfies
ǫ
|tn − s| < (N + 1)(2M )
4(N + 1)M
n
ǫ X
+ pnk < ǫ
2
k=N +1
′
whenever n > N . This implies tn → s as n → ∞ and completes the proof for
the sufficiency.
Classical Methods from Infinitesimal Calculus 11

Example 1.1.10 We now give some examples of applications to Theorem

1.1.9. The first one is
1
lim (p0 p1 p2 · · · pn ) n+1 = p, (1.9)
n→∞

where (pk )k≥0 is a positive sequence that approaches to p as n → ∞. Indeed,

the left-hand side of (1.9) can be written as
n
!
1 X
lim exp log pk .
n→∞ n+1
k=0
n Pn o
1
Sequence n+1 k=0 log pk can be considered as the transfered matrix
n≥0
from (pk ) with respect to the transformationP matrix [pnk = 1/(n + 1)]. Since
1 n
pn → p, from Theorem 1.1.9, we have n+1 k=0 log pk → log p as n → ∞.
Thus, (1.9) holds.
By setting
 1  2  n
2 3 n+1
p0 = 1, p1 = , p2 = , · · · , pn ,···
1 2 n
in (1.9), we obtain
  n+1
1  n
(n + 1)n+1 1
lim = lim 1+ = e, (1.10)
n→∞ (n + 1)! n→∞ n
  n1
nn
i.e., lim = e.
n→∞ n!
P
Corollary 1.1.11 Let (an ) and (bn ) be two sequences with bn > 0, n≥0 bn =
∞, and an /bn → s (n → ∞). Then
a0 + a1 + a2 + · · · + an
lim = s. (1.11)
n→∞ b0 + b1 + · · · + bn
Proof. (1.11) can be proved from Theorem 1.1.9 by setting sn = an /bn , pnk =
bk /(b0 + b1 + · · · + bn ), and tn = (a0 + a1 + · · · + an )/(b0 + b1 + · · · + bn ).

Example 1.1.12 In (1.11), if an = (n+1)α−1 (α > 0) and bn = (n+1)α −nα ,

then we have
 
1α−1 + 2α−1 + · · · + nα−1 nα−1 1
lim = lim = .
n→∞ nα n→∞ (n + 1)α − nα α
Similarly, if a positive sequence (pn )n≥0 satisfies
pn
lim = 0,
n→∞ p0 + p1 + · · · + pn
12 Methods for the Summation of Series

then from an assumption lim sn = s, we obtain

n→∞

s0 pn + s1 pn−1 + s2 pn−2 + · · · + sn p0
lim = s. (1.12)
n→∞ p0 + p1 + · · · + pn
P∞ Pn
Let pn > 0, k=1 pk = ∞, and let k=1 pk = Pn with pn /Pn → 0 as
n → ∞. Then
p1 P1−1 + p2 P2−1 + · · · + pn Pn−1
lim = 1. (1.13)
n→∞ log Pn
In fact, from (1.11) the left-hand limit of (1.13) can be written as

p1 P1−1 + p2 P2−1 + · · · + pn Pn−1

lim
n→∞ log P1 + (log P2 − log P1 ) + (log P3 − log P2 ) + · · ·
+(log Pn − log Pn−1 )
pn Pn−1 p P −1
= lim = lim n n  = 1.
n→∞ log Pn − log Pn−1 n→∞
log 1 + Pn Pn−1

If pn = 1 (n = 1, 2, . . .), then (1.13) implies

1 1 1
1+ + + · · · + ∼ log n
2 3 n
as n → ∞.
The final example is related to positive sequences (pn ) and (qn ) satisfying
p1 + p2 + · · · pn q1 + q2 + · · · qn
lim = a, lim = b,
n→∞ npn n→∞ nqn
where 0 < a, b < ∞. Then
p1 q1 + 2p2 q2 + 3p3 q3 + · · · + npn qn ab
lim = . (1.14)
n→∞ n2 pn qn a+b
P∞ P∞
To prove (1.14), we first observe that k=1 pk = ∞ and k=1 qk = ∞.
Otherwise npn → 0 and nqn → 0 as n → ∞, which imply a = b = ∞ that
contradict to the assumption. Denote

an = Pn Qn − Pn−1 Qn−1 , bn = npn qn ,

Pn Pn
where Pn = k=1 pk and Qn = k=1 qk . Thus
an Pn Qn 1
= + − →a+b (n → ∞).
bn npn nqn n
Therefore, from Corollary 1.1.11 we obtain
a1 + a2 + · · · + an Pn Qn
lim = lim Pn = a + b,
n→∞ b1 + b2 + · · · + bn n→∞
k=1 kpk qk
Classical Methods from Infinitesimal Calculus 13

which yields
Pn  Pn  
k=1 kpk qk k=1 kpk qk Pn Qn ab
= → ,
n2 pn qn Pn Qn n2 pn qn a+b
as n → ∞.

Corollary 1.1.13 Suppose two positive sequences (pn ) and (qn ) satisfy
pn qn
lim = 0, lim = 0.
n→∞ p0 + p1 + · · · + pn n→∞ q0 + q1 + · · · + qn
P
And denote rn = nk=0 pk qn−k (n = 0, 1, . . .). Then
rn
lim = 0. (1.15)
n→∞ r0 + r1 + · · · + rn
Proof. Denote
n
X n
X n
X
Pn = pk , Qn = qk , Rn = rk .
k=0 k=0 k=0

Then
rn p0 qn + p1 qn−1 + · · · + pn q0
=
Rn p0 Qn + p1 Qn−1 + · · · + pn Q0
     
q0 q1 qn
= pn0 + pn1 + · · · + pnn ,
Q0 Q1 Qn
where
pn−j Qj pn−j
pnj = ≤ →0
p0 Qn + p1 Qn−1 + · · · pn Q0 p0 + p1 + · · · + pn−j

as n → ∞, and pn0 + pn1 + · · · pnn = 1. Hence, (1.15) is simply a special case

of the conclusion of Theorem 1.1.9.

Example 1.1.14 Suppose that sequences (pn ) and (qn ) are defined as Corol-
lary 1.1.13. Let (sn ) be any sequence. Then the existence of limits
s0 pn + s1 pn−1 + · · · + sn p0
lim = p̄
n→∞ p0 + p1 + · · · pn
and
s0 qn + s1 qn−1 + · · · + sn q0
lim = q̄
n→∞ q0 + q1 + · · · qn
implies p̄ = q̄. Indeed, let (rn ) be the sequence defined in Corollary 1.1.13, and
let
s0 rn + s1 rn−1 + · · · + sn r0
τn := .
r0 + r1 + · · · rn
14 Methods for the Summation of Series

Denote
s0 pn + s1 pn−1 + · · · + sn p0 s0 qn + s1 qn−1 + · · · + sn q0
p̄n := , and q̄n := .
p0 + p1 + · · · pn q0 + q1 + · · · qn
From Corollary 1.1.13 we have
pn Q0 q̄0 + pn−1 Q1 q̄1 + · · · + p0 Qn q̄n
τn =
p0 Qn + p1 Qn−1 + · · · pn Q0
qn P0 p̄0 + qn−1 P1 p̄1 + · · · + q0 Pn p̄n
= .
q0 Pn + q1 Pn−1 + · · · qn P0

Thus, the same argument in the proof of Corollary 1.1.13 yields lim τn =
n→∞
lim p̄n = lim q̄n . Consequently, p̄ = q̄. It is interesting to see that this
n→∞ n→∞
conclusion holds even lim sn does not exist.
n→∞

Corollary 1.1.15 If σ > 0, then the convergence of Dirichlet series

a1 1−σ + a2 2−σ + · · · + an n−σ + · · ·

implies
(a1 + a2 + · · · + an )n−σ → 0
as n → ∞.

Proof. Denote

tn = (a1 + a2 + · · · + an )n−σ ,
sn = a1 1−σ + a2 2−σ + · · · + an n−σ .

Then

tn − n−σ (n + 1)σ (sn − s)

" n #
1 X σ
= (k − (k + 1)σ )(sk − s) + s .
nσ
k=1

Hence, from Theorem 1.1.9, we obtain the right-hand side of the above equa-
tion tends to zero as n → ∞, which implies tn → 0 (n → ∞).

Corollary 1.1.15 can be proved by using Abel’s summation by part method

(see 1.2.2).
Remark 1.1.16 We can reduce the request of the row sums of a transfor-
mation matrix to be uniformly bounded by a constant K > 0. This matrix
Classical Methods from Infinitesimal Calculus 15

is called the transformation matrix with respect to K. In this case, the conse-
quence of Theorem 1.1.9 is changed to be that lim tn = Ks when lim sn = s.
n→∞ n→∞
Furthermore, if a lower triangular matrix [pnk ]n≥k≥0 has entries satisfying
n
X
pnk = Pn → 1
k=0

as n → ∞ and pnk ≥ 0, then this lower triangular matrix can be used to con-
struct a sequence transformation. Theorem 1.1.9 still holds for those transfor-
mation matrices. We leave the proof of those claims as Exercise 1.8.
Definition 1.1.17 A sequence (sn ) is termed a null sequence if for any given
ǫ > 0, there exists an integer N ≡ N (ǫ) such that n > N implies |sn | < ǫ.
From Theorem 1.1.9 and Remark 1.1.16, we immediately have
Theorem 1.1.18 Let (sn ) be a null sequence, and let (tn ) the transformed
sequence of (sn ) using a transformation matrix with respect to constant K > 0.
Then (tn ) is also a null sequence if for every fixed m
lim pnm = 0.
n→∞

Proof. For any given ǫ > 0, there exists N ≡ N (ǫ) such that for every n > N ,
|sn | < ǫ/(2K). Then for that n,
N
X ǫ
|tn | < pnk sk + .
2
k=0

By lim pnm = 0 we may choose N0 > N so that for every n > N0 ,

n→∞

N
X ǫ
pnk sk < .
2
k=0

Therefore, we have shown that |tn | < ǫ for these n′ s, which completes the
proof of the theorem.

Theorem 1.1.9 can be extended to the following theorem.

Theorem
P∞ 1.1.19 Let positive infinite matrix P = [pnk ]0<n,k<∞ satisfies
k=0 p nk = 1. And the sequence
X
tn = pnk sk
k≥0

is called the transformation sequence of (sn ) with respect to the transformation

matrix P . Then lim sn = s implies lim tn = s if and only if lim pnk = 0
n→∞ n→∞ n→∞
for all k = 0, 1, 2, . . ..
16 Methods for the Summation of Series

The proof of Theorem 1.1.19 is similar to the proof of Theorem 1.1.9. We

leave it as an exercise for the interested reader (see Exercise 1.10).
P
Example 1.1.20 (i) If sequence k≥1 kck converges, then the sequence (tn )
defined by X
tn = (k + 1)cn+k
k≥0

converges to 0. Actually, the limit P

can be proved by using Theorem 1.1.19 and
the sequence (sn ) defined by sn = k≥0 (n + k)cn+k → 0 (n → ∞). Hence,
   
1 2 1 3 2
tn = s n + − sn+1 + − sn+2 + · · · .
n n+1 n n+2 n+1
From Theorem 1.1.19 and notingP limn→∞ sn = 0, we have limn→∞ tn = 0.
(ii) Let power series f (x) = k≥0 ak xk converge at x = 1, and let 0 <
α < 1. Then the power series

f ′ (α) f ′′ (α) 2 f (n) (α) n

f (α) + h+ h + ··· + h + ··· (1.16)
1! 2! n!
converges to f (1) at h = 1 − α. To prove (1.16), we denote

f (n) (α)
a0 + a1 + · · · + an = sn , = bn ,
n!
b0 + b1 (1 − α) + b2 (1 − α) + · · · + bn (1 − α)n = tn .
2

Hence, for |y| < 1 − α,

b0 + b1 y + b2 y 2 + · · · + bn y n + · · ·
= a0 + a1 (α + y) + · · · + an (α + y)n + · · · .

Consequently,
X X (1 − α)n+1 X
(1 − α)n−k y k bj y j = aj (α + y)j
1 − (α + y)
k≥0 j≥1 j≥0
X
= (1 − α)n+1 sj (α + y)j .
j≥0

n
Comparing the coefficients of y on the leftmost and rightmost sides of the
above equation yields
X n + k 
n+1
tn = (1 − α) αk sn+k .
k
k≥0

Noting that the sum of the coefficients of the series is 1, from Theorem 1.1.19,
we obtain
lim tn = lim sn = f (1).
n→∞ n→∞
Classical Methods from Infinitesimal Calculus 17

More convergence of the transformation series will be described in Sub-

section 1.3.1. At the end of this section, we discuss the limits of series they
are equivalent to the existence of integrals. We start from the limits of some
sequences and their evaluation using suitable integrals. For example, from the
definition of Riemann integrals, we have
n
X X n Z 1
1 1 1 1
lim = lim = dx = log 2.
n→∞ n + k n→∞ 1 + k/n n 0 1+x
k=1 k=1

Similarly,
n
X n π
lim = ,
n→∞ n2 + k 2 4
k=1
Xn
1 kπ 2
lim sin = ,
n→∞ n n π
k=1
Xn
1 kπ 4
lim sec2 = ,
n→∞ n 4n π
k=0
Xn
kα 1
lim = (α > −1).
n→∞ nα+1 α+1
k=1

Since !
n
1√n 1X k
n! = exp log ,
n n n
k=1

we immediately know limit

Z 
1√
n
1
1
lim n! = exp log xdx = .
n→∞ n 0 e

The following limit belongs to Pólya:

n   h n i
1X 2n
lim −2 = 2 log 2 − 1, (1.17)
n→∞ n k k
k=1

where [x] denotes the largest integer ≤ x. Obviously,

  hni  2   
2n 1
0≤ −2 = −2 ≤ 1.
k k k/n k/n

Thus,
n   h n i Z 1  2   
1X 2n 1
lim −2 = −2 dx. (1.18)
n→∞ n k k 0+ x x
k=1
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