Sequences and seriesjeanmariedufour.github.io/ResE/Dufour_1992_C_TS_SequencesSeries… · Sequences...

Sequences and series ∗

Jean-Marie Dufour †

McGill University

First version: March 1992Revised: January 2002, October 2016

This version: October 2016Compiled: January 10, 2017, 15:36

∗ This work was supported by the William Dow Chair in Political Economy (McGill University), the Bank of Canada (Research Fellowship), the Toulouse School of Economics (Pierre-de-Fermat Chairof excellence), the Universitad Carlos III de Madrid (Banco Santander de Madrid Chair of excellence), a Guggenheim Fellowship, a Konrad-Adenauer Fellowship (Alexander-von-Humboldt Foundation,Germany), the Canadian Network of Centres of Excellence [program on Mathematics of Information Technology and Complex Systems (MITACS)], the Natural Sciences and Engineering Research Councilof Canada, the Social Sciences and Humanities Research Council of Canada, and the Fonds de recherche sur la société et la culture (Québec).

† William Dow Professor of Economics, McGill University, Centre interuniversitaire de recherche en analyse des organisations (CIRANO), and Centre interuniversitaire de recherche en économiequantitative (CIREQ). Mailing address: Department of Economics, McGill University, Leacock Building, Room 414, 855 Sherbrooke Street West, Montréal, Québec H3A 2T7, Canada. TEL: (1) 514 3984400 ext. 09156; FAX: (1) 514 398 4800; e-mail: [email protected] . Web page: http://www.jeanmariedufour.com

Contents

List of Definitions, Assumptions, Propositions and Theorems v

1. Definitions and notation 1

2. Convergence of sequences 8

3. Convergence of series 12

4. Convergence of transformed series 21

5. Uniform convergence 26

6. Proofs and references 29

i

List of Tables

List of Figures

List of Definitions, Assumptions, Propositions and Theorems

1.3 Definition : Bounded set in R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Definition : Supremum and infimum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 Definition : Bounded set in C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.6 Definition : Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.8 Definition : Subsequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.9 Definition : Limit of a complex sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.11 Definition : Convergence in a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.12 Definition : Convergence in the sense of Cauchy . . . . . . . . . . . . . . . . . . . . . . . . . 41.13 Definition : Infinite limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.14 Definition : Monotonic sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.15 Definition : Upper and lower limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.17 Definition : Accumulation point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.18 Definition : Partial sum and series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.20 Definition : Absolute and conditional convergence . . . . . . . . . . . . . . . . . . . . . . . . 61.21 Definition : Two-sided sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.22 Definition : Double sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.23 Definition : Limit of a complex double sequence . . . . . . . . . . . . . . . . . . . . . . . . . 7

ii

2.1 Theorem : Convergence criterion for complex sequences . . . . . . . . . . . . . . . . . . . . 82.2 Theorem : Properties of convergent sequences . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Theorem : Convergence of transformed sequences . . . . . . . . . . . . . . . . . . . . . . . . 92.4 Theorem : Convergence of Cauchy sequences . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5 Theorem : Convergence of real sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.6 Theorem : Limits of important special sequences . . . . . . . . . . . . . . . . . . . . . . . . 113.1 Theorem : Cauchy criterion for convergence of a series . . . . . . . . . . . . . . . . . . . . . 123.2 Proposition : Alternative forms of the Cauchy criterion for series . . . . . . . . . . . . . . . . 123.3 Theorem : Necessary conditions for convergence of a series . . . . . . . . . . . . . . . . . . . 123.4 Corollary : Divergence criterion for a series . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.5 Theorem : Characterization of absolute convergence . . . . . . . . . . . . . . . . . . . . . . . 133.7 Theorem : Criterion for absolute convergence of a series . . . . . . . . . . . . . . . . . . . . 133.8 Corollary : Divergence criterion for a series . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.9 Theorem : Comparison criterion for convergence of a series . . . . . . . . . . . . . . . . . . . 133.10 Theorem : Convergence of a series of nonnegative numbers . . . . . . . . . . . . . . . . . . . 143.11 Proposition : Convergence of special series . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.12 Theorem : Root convergence criterion (Cauchy) . . . . . . . . . . . . . . . . . . . . . . . . . 153.13 Theorem : Ratio convergence criterion (d’Alembert) . . . . . . . . . . . . . . . . . . . . . . 153.14 Theorem : Relation between the root and ratio convergence tests . . . . . . . . . . . . . . . . 163.15 Theorem : Cauchy criterion for convergence of a series . . . . . . . . . . . . . . . . . . . . . 163.16 Theorem : Gauss criterion for convergence of a series . . . . . . . . . . . . . . . . . . . . . . 17

iii

3.17 Theorem : Integral criterion for convergence of a series . . . . . . . . . . . . . . . . . . . . . 173.18 Theorem : Dirichlet criterion for convergence of a series of products . . . . . . . . . . . . . . 183.19 Corollary : Alternating series convergence criterion (Leibniz) . . . . . . . . . . . . . . . . . . 183.20 Theorem : Abel criterion for convergence of a series of products . . . . . . . . . . . . . . . . 193.22 Theorem : Abel-Dini criterion for convergence of a series of ratios . . . . . . . . . . . . . . . 193.23 Theorem : Landau criterion for convergence of a series of products . . . . . . . . . . . . . . . 193.25 Theorem : Convergence of an arithmetically weighted mean . . . . . . . . . . . . . . . . . . . 203.26 Theorem : Cesaro convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.1 Theorem : Convergence of linearly transformed series . . . . . . . . . . . . . . . . . . . . . . 214.2 Definition : Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.4 Theorem : Sufficient condition for convergence of the product of two series (Cauchy-Mertens) . 224.5 Theorem : Limit of the product of two series (Abel) . . . . . . . . . . . . . . . . . . . . . . . 224.6 Theorem : Necessary and sufficient condition for convergence of the product of two series . . . 224.7 Theorem : Aggregation of terms in a series . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.8 Definition : Rearrangement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.9 Definition : Commutatively convergent series . . . . . . . . . . . . . . . . . . . . . . . . . . 234.10 Theorem : Rearrangement of an absolutely convergent series (Dirichlet) . . . . . . . . . . . . 244.11 Theorem : Rearrangement of a conditionnally convergent series (Riemann) . . . . . . . . . . . 244.12 Theorem : Equivalence between absolute and commutative convergence . . . . . . . . . . . . 244.13 Theorem : Condition for double series commutativity . . . . . . . . . . . . . . . . . . . . . . 255.2 Definition : Uniform convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

iv

5.4 Theorem : Cauchy criterion for uniform convergence . . . . . . . . . . . . . . . . . . . . . . 275.5 Theorem : Supremum criterion for uniform convergence . . . . . . . . . . . . . . . . . . . . 275.6 Theorem : Weierstrass uniform convergence criterion . . . . . . . . . . . . . . . . . . . . . . 275.7 Theorem : Uniform convergence and continuity . . . . . . . . . . . . . . . . . . . . . . . . . 275.9 Theorem : Conditions of uniform convergence (Dini) . . . . . . . . . . . . . . . . . . . . . . 285.10 Theorem : Uniform convergence and differentiation of functions of real variables . . . . . . . . 285.11 Theorem : Uniform convergence and differentiation of functions of complex variables . . . . . 28

v

1. DEFINITIONS AND NOTATION 1

1. Definitions and notation

1.1 Notation We shall use the following notation:

(1) iff : if and only if;

(2) ⇔ : if and only if;

(3) ∞ : infinity ;

(4) Ac : complement of the set A;

(5) ⇒ : implies ;

(6) ∼ : is distributed like;

(7) ≡ : equal by definition;

(8) C : set of complex numbers;

(9) R : real numbers;

(10) Z : integers;

(11) N0 = {0, 1, 2, ...} : nonnegative integers;

(12) N = {1, 2, 3, ...} : positive integers;

(13) R : extended real numbers :R = R∪{+∞}∪{−∞} .


1.2 Notation 1.3 Definition BOUNDED SET IN R. Let E ⊆ R. If there is an element y ∈ R such that x ≤ y,∀x ∈ R, we say that E has an upper bound (or is bounded from above). If there is an element z ∈ R such thatx ≥ z, ∀x ∈ E, we say that E has a lower bound (or is bounded from below). If E has both upper and lowerbounds, we say that E is bounded.

1.4 Definition SUPREMUM AND INFIMUM. Let E ⊆ R. sup(E) is the smallest element of R such thatx ≤ sup(E), ∀x ∈ E; inf(E) is the largest element of R such that inf(E) ≤ x, ∀x ∈ E.

1.5 Definition BOUNDED SET IN C. Let E ⊆ C. If there is a real number M and a complex number z0 suchthat |z− z0| < M for all z ∈ E, we say the set E is bounded.

1.6 Definition SEQUENCE. Let E be a set. A sequence in E is function f (n) = an which associates to eachelement n ∈ N an element an ∈ E. The sequence is usually denoted by the ordered set of the values of f (n):

{a1,a2, . . .} ≡ {an}∞n=1 ≡ {an}

or(a1,a2, . . .) ≡ (an)

∞n=1 ≡ (an) .

If E = C, the sequence is complex. If E = R, the sequence is real. To indicate that all the elements of thesequence {an} are in E, we write {an} ⊆ E.


1.7 Remark Let m ∈Z and Im = {n ∈ Z : n ≥ m}. A function f (n) = bn which maps every element n ∈ Im toan element an ∈ E can be viewed as a sequence in E on defining an = bm+n−1, n = 1,2, . . . Such a sequenceis usually denoted

{bm,bm+1, . . .} ≡ {bn}∞n=m .

Similarly, if Im = {n ∈ Z : n ≤ m}, we can define an = bm−n+1, n = 1,2, . . . . In this case, the sequence canbe denoted as

{...,bm−1,bm} ≡ {bn}mn=−∞ .

1.8 Definition SUBSEQUENCE. Let E be a set, {an}∞n=1 ⊆ E, and {nk}∞

k=1 a sequence of positive integerssuch that n1 < n2 < · · · . The sequence

{

ank

}∞k=1 is a subsequence of {an}∞

n=1 .

1.9 Definition LIMIT OF A COMPLEX SEQUENCE. Let a ∈ C and {an} ⊆ C. The sequence {an} converges

to a iff for any real number ε > 0, there is an integer N such that n ≥ N implies |an−a| < ε . In this case,we write an → a, or

limn→∞

an = a ,

and a is called the limit of {an}. If there is a number a ∈ C such that an → a, we say that the sequence {an}converges (or converges in C). If the sequence does not converge, we say it diverges.

1.10 Remark When there is no ambiguity, we can also write lim an instead of limn→∞ an.

1.11 Definition CONVERGENCE IN A SET. Let E ⊆ C and {an} ⊆ E. If there exists an element a ∈ E suchthat an → a, we say that {an} converges in E.


1.12 Definition CONVERGENCE IN THE SENSE OF CAUCHY. Let {an}⊆C. The sequence {an} converges

in the Cauchy sense iff for any ε > 0, there exists an integer N such that m≥N and n≥N imply |am−an|< ε .A sequence which converges in the Cauchy sense is called a Cauchy sequence.

1.13 Definition INFINITE LIMITS. Let {an} ⊆ R. We say that the sequence {an} diverges to ∞ iff for anyreal number M there exists an integer N such that n ≥ N implies an ≥ M. In this case, we write an → ∞ or

limn→∞

an = ∞ .

Similarly, we say the sequence {an} diverges to −∞ iff for any real number M there is an integer N such thatn ≥ N implies an ≤ M. In this case, we write an →−∞ or

limn→∞

an = −∞ .

We also wrote +∞ instead of ∞.

1.14 Definition MONOTONIC SEQUENCE. Let {an} ⊆ R. If an ≤ an+1, for all n ∈ N, we say that thesequence {an} is monotonically increasing (or monotonic increasing) . If an ≥ an+1 for all n ∈ N, we saythe sequence {an} is monotonically decreasing (monotonic decreasing). If {an} is monotonically increasingand an → a, we write an ↑ a. If {an} is monotonically decreasing and an → a, we write an ↓ a.


1.15 Definition UPPER AND LOWER LIMITS. Let {an} ⊆ R. The upper limit of the sequence {an} isdefined by

limsupn→∞

an = infN≥1

{

supn≥N

an

}

≡ inf{sup{an : n ≥ N} : N ≥ 1} .

The lower limit of the sequence {an}is defined by

liminfn→∞

an = supN≥1

{

infn≥N

an

}

≡ sup{inf{an : n ≥ N} : N ≥ 1} .

We also write lim instead of limsup, and lim instead of liminf.

1.16 Remark The upper and lower limits of the sequence {an} ⊆ R always exist in R.

1.17 Definition ACCUMULATION POINT. Let {an} ⊆ C and a ∈ C. a is an accumulation point of {an} ifffor any real number ε > 0, the inequality |an−a| < ε is satisfied for an infinity of elements of the sequence{an} .


1.18 Definition PARTIAL SUM AND SERIES. Let {an} ⊆ C and SN = ∑Nn=1 an. We call {SN}∞

N=1 thesequence of partial sums associated with {an}. The symbol ∑∞

n=1 an represents the series associated with{an}. If limN→∞ SN = S where S ∈ C, we say the series ∑∞

n=1 an converges (or converges to S) and we write∞

∑n=1

an = S .

If the series ∑∞n=1 an does not converge, we say it diverges.

1.19 Remark If we consider a sequence of the form {an}∞n=m where m ∈ Z, we say that the series ∑∞

n=m an

converges to S if limN→∞ SN = S, where SN = ∑N+(m−1)n=m an. Similarly, for a sequence of the form {an}m

n=−∞,where m ∈ Z, we say that the series ∑m

n=−∞ an converges to S if limN→∞ SN = S, where SN = ∑m+1−Nn=m an.

1.20 Definition ABSOLUTE AND CONDITIONAL CONVERGENCE. Let {an} ⊆ C. If the series ∑∞n=1 |an|

converges, we say that the series ∑∞n=1 an converges absolutely. If ∑∞

n=1 an converges, but ∑∞n=1 |an| does not

converge, we say that ∑∞n=1 an converges conditionally.

1.21 Definition TWO-SIDED SEQUENCE. Let {an}∞n=0 and {an}−1

n=−∞ be two sequences of complex num-bers. If the series ∑∞

n=0 an converges to S1 ∈ C and if the series ∑−1n=−∞ an converges to S2 ∈ C, we say that the

two-sided series ∑∞n=−∞ an converges to S1 +S2.


1.22 Definition DOUBLE SEQUENCE. A double sequence in E is a function f (m, n) = amn which mapseach pair (m,n) ∈ N2 to an element amn ∈ E. We usually denote the double sequence by

{amn}∞m,n=1 ≡ {amn} .

To indicate that all the elements of the double sequence {amn} are in E, we write {amn} ⊆ E.

1.23 Definition LIMIT OF A COMPLEX DOUBLE SEQUENCE. Let a ∈ C and {amn} ⊆ C. The doublesequence {amn} converges to a when m, n → ∞ iff for any real number ε > 0, there is an integer N such thatm, n ≥ N implies |amn−a| < ε . In this case, we write amn −→

m,n→∞a, or

limm,n→∞

amn = a ,

and a is called the limit of {amn} when m, n → ∞.

1.24 Remark For double sequences, we can consider several different limits: limm→∞ amn, limn→∞ amn,

limm→∞ [limn→∞ amn] , limn→∞ [limm→∞ amn] . In general, these limits are not equal. Even if

limm→∞

amn ≡ bn , limn→∞

amn = cm

exist, we can havelimn→∞

[

limm→∞

amn

]

≡ limn→∞

bn 6= limm→∞

cm ≡ limm→∞

[

limn→∞

amn

]

.

2. CONVERGENCE OF SEQUENCES 8

2. Convergence of sequences

2.1 Theorem CONVERGENCE CRITERION FOR COMPLEX SEQUENCES. Let {cn} ⊆ C, where cn = an +

i bn, an ∈ R, bn ∈ R, for all n, and i =√−1. Then the sequence {cn} converges iff each one of the sequences

{an} and {bn} converges. Furthermore, if {cn} converges, we have:

limn→∞

cn =(

limn→∞

an

)

+ i(

limn→∞

bn

)

.

2.2 Theorem PROPERTIES OF CONVERGENT SEQUENCES. Let {an} ⊆ C, a ∈ C and a′ ∈ C.

(a) Limit unicity. If an → a and an → a′, then a = a′.

(b) Boundedness of convergent sequences. If the sequence {an} converges, then the set {a1, a2, ...} isbounded.

(c) Convergence of bounded sequences (Bolzano-Weierstrass). If the sequence {an} is bounded, then{an} contains a convergent subsequence {ank

}. In other words, a bounded sequence {an} has at leastone accumulation point.

(d) Convergence of subsequences. {an} converges to a ⇔ each subsequence {ank} of {an} converges to

a ⇔ each subsequence {ank} of {an} contains another subsequence {amk

} which converges to a.

(e) Accumulation and convergence. If the sequence {an} is bounded and has exactly one accumulationpoint a, then an → a. If the sequence {an} has no finite accumulation point or has several, then itdiverges.


2.3 Theorem CONVERGENCE OF TRANSFORMED SEQUENCES. Let {an} ⊆ C and {bn} ⊆ C two se-quences such that

limn→∞

an = a, limn→∞

bn = b

where a, b ∈ C. Then

(a) limn→∞ (an +bn) = a+b ;

(b) limn→∞ (c an) = c a, limn→∞ (an + c) = a+ c, ∀c ∈ C ;

(c) limn→∞ (anbn) = a b ;

(d) lim(an/bn) = a/b, provided b 6= 0 ;

(e) limn→∞ g(an) = g(a) for any function g : C→ C continuous at x = a.

2.4 Theorem CONVERGENCE OF CAUCHY SEQUENCES. Let {an} ⊆ C.

(a) If the sequence {an} converges, then {an} converges in the Cauchy sense.

(b) Completeness. If the sequence {an} converges in the Cauchy sense, then the sequence {an} converges.


2.5 Theorem CONVERGENCE OF REAL SEQUENCES. Let {an} ⊆ R, {bn} ⊆ R, a ∈ R and b ∈ R.

(a) liminfn→∞

an ≤ limsupn→∞

an.

(b) limn→∞

an = a ⇔ liminfn→∞

an = limsupn→∞

an = a.

(c) If an ≤ bn for n ≥ N, then

liminfn→∞

an ≤ liminfn→∞

bn ,

limsupn→∞

an ≤ limsupn→∞

bn .

(d) If an ≤ bn for n ≥ N and if {an} and {bn} are convergent sequences, then

limn→∞

an ≤ limn→∞

bn .

(e) If the sequence {an} is monotonically increasing, then

{an} converges in R or limn→∞

an = ∞ .

( f ) If the sequence {an} is monotonically decreasing, then

{an} converges in R or limn→∞

an = −∞ .

(g) If the sequence {an} is monotonic (increasing or decreasing) and bounded, then {an} converges in R.


2.6 Theorem LIMITS OF IMPORTANT SPECIAL SEQUENCES. Let p and α be real numbers and x a complexnumber.

(a) If p > 0, limn→∞

1np = 0 .

(b) If p > 0, limn→∞

p1/n = 1 .

(c) limn→∞

n1/n = 1 .

(d) If p > 0, limn→∞

nα

(1+p)n = 0 .

(e) If |x| < 1, limn→∞

xn = 0 .

( f ) If b > 0 and b 6= 1, limn→∞

[logb(n)/n] = 0 .

3. CONVERGENCE OF SERIES 12

3. Convergence of series

3.1 Theorem CAUCHY CRITERION FOR CONVERGENCE OF A SERIES. Let {an} ⊆ C. The series ∑∞n=1 an

converges iff, for any ε > 0, there is an integer N such that n ≥ m ≥ N implies |∑nk=m ak| < ε.

3.2 Proposition ALTERNATIVE FORMS OF THE CAUCHY CRITERION FOR SERIES. Let {an} ⊆ C. Theseries ∑∞

n=1 an converges

⇔ for any ε > 0, there is an integer N such that n ≥ N implies∣

∣∑n+pk=n+1 an

∣

∣< ε for any p ≥ 1

⇔ for any ε > 0, there is an integer N such that n ≥ N implies supp≥1

∣

∣∑n+pk=n+1 an

∣

∣< ε

⇔ limn→∞

[

supp≥1

∣

∣∑n+pk=n+1 an

∣

∣

]

= 0 .

3.3 Theorem NECESSARY CONDITIONS FOR CONVERGENCE OF A SERIES. Let {an} ⊆ C.

(a) If the series ∑∞n=1 an converges, then lim

n→∞an = 0 .

(b) If the series ∑∞n=1 |an| converges and if |an+1| ≤ |an| for n ≥ N, then lim

n→∞(nan) = 0 .

3.4 Corollary DIVERGENCE CRITERION FOR A SERIES. Let {an} ⊆C and c > 0. If |an| ≥ c for an infinitenumber values of n, then the series ∑∞

n=1 an diverges.


3.5 Theorem CHARACTERIZATION OF ABSOLUTE CONVERGENCE. Let {an} ⊆ C. The series ∑∞n=1 an

does not converge absolutely ⇔ ∑∞n=1 |an| diverges ⇔ ∑∞

n=1 |an| = ∞ .

3.6 Remark To indicate that the series ∑∞n=1 an converges absolutely, we can write ∑∞

n=1 |an| < ∞.

3.7 Theorem CRITERION FOR ABSOLUTE CONVERGENCE OF A SERIES. Let {an} ⊆ C. If the series∑∞

n=1 an converges absolutely, then ∑∞n=1 an converges.

3.8 Corollary DIVERGENCE CRITERION FOR A SERIES. Let {an} ⊆C. If the series ∑∞n=1 an diverges, then

∑∞n=1 |an| = ∞ .

3.9 Theorem COMPARISON CRITERION FOR CONVERGENCE OF A SERIES. Let {an} ⊆ C, {cn} ⊆ R and{dn} ⊆ R.

(a) If |an| ≤ cn for n ≥ n0, where n0 is a given integer, and if the series ∑∞n=1 cn converges, then the series

∑∞n=1 an converges absolutely.

(b) If |an| ≥ dn ≥ 0 for n ≥ n0, where n0 is a given integer, and if ∑∞n=1 dn diverges, then ∑∞

n=1 |an| = ∞ but∑∞

n=1 an can converge or diverge.


3.10 Theorem CONVERGENCE OF A SERIES OF NONNEGATIVE NUMBERS. Let {an} ⊆ R where an ≥ 0for all n.

(a) The series ∑∞n=1 an converges iff the sequence of partial sums {∑N

n=1 an}∞N=1 is bounded.

(b) Cauchy’s condensation criterion. If the sequence {an} is monotonically decreasing (a1 ≥ a2 ≥ a3 ≥...), the series ∑∞

n=1 an converges iff the series∞

∑k=0

2ka2k = a1 +2a2 +4a4 +8 a8 + ... (3.1)

converges.

3.11 Proposition CONVERGENCE OF SPECIAL SERIES. Let x a complex number and p a real number.

(a) Geometric series. If |x| < 1 ,∞

∑n=0

xn =1

1− x

where 00 ≡ 1. If |x| ≥ 1, the series ∑∞n=0 xn diverges.

(b) The series ∑∞n=1 1/np converges if p > 1 and diverges if p ≤ 1.

(c) The series ∑∞n=2

1n(logn)p converges if p > 1 and diverges if p ≤ 1.

(d) e = ∑∞n=0

1n! = lim

n→∞

(

1+ 1n

)n.


3.12 Theorem ROOT CONVERGENCE CRITERION (CAUCHY). Let {an} ⊆ C and α = limsupn→∞

|an|1/n .

(a) If α < 1, the series ∑∞n=1 an converges absolutely.

(b) If α > 1, ∑∞n=1 an diverges.

(c) If |an|1/n ≤ δ < 1 for n ≥ n0, where n0 is a given integer, the series ∑∞n=1 an converges absolutely.

(d) If |an|1/n ≥ 1 for an infinite number of values of n, ∑∞n=1 an diverges.

(e) If none of the preceding conditions holds, we can find cases where ∑∞n=1 an converges and cases where

∑∞n=1 an diverges.

3.13 Theorem RATIO CONVERGENCE CRITERION (D’ALEMBERT). Let {an} ⊆ C and 0/0 ≡ 0.

(a) If limsupn→∞

∣

∣

∣

an+1an

∣

∣

∣< 1, the series ∑∞

n=1 an converges absolutely.

(b) If∣

∣

∣

an+1an

∣

∣

∣≤ ε < 1 for n ≥ n0, where n0 is a given integer, the series ∑∞

n=1 an converges absolutely.

(c) If∣

∣

∣

an+1an

∣

∣

∣≥ 1 for n ≥ n0, where n0 is a given integer, the series ∑∞

n=1 an diverges.

(d) If liminfn→∞

∣

∣

∣

an+1an

∣

∣

∣> 1, the series ∑∞

n=1 an diverges.

(e) If liminfn→∞

∣

∣

∣

an+1an

∣

∣

∣≤ 1 ≤ limsup

n→∞

∣

∣

∣

an+1an

∣

∣

∣, the series ∑∞



3.14 Theorem RELATION BETWEEN THE ROOT AND RATIO CONVERGENCE TESTS. Let {an} ⊆ C asequence such that an 6= 0 for n ≥ n0, where n0 is a given integer. Then

liminfn→∞

∣

∣

∣

∣

an+1

an

∣

∣

∣

∣

≤ liminfn→∞

|an|1/n ≤ limsupn→∞

|an|1/n ≤ limsupn→∞

∣

∣

∣

∣

an+1

an

∣

∣

∣

∣

.

If we define 0/0 ≡ 0 and |x/0| = ∞ for x 6= 0, the above inequalities hold for any sequence {an} ⊆ C.

3.15 Theorem CAUCHY CRITERION FOR CONVERGENCE OF A SERIES. Let {an} ⊆ C and

L = liminfn→∞

n

(

1−∣

∣

∣

∣

an+1

an

∣

∣

∣

∣

)

, U = limsupn→∞

n

(

1−∣

∣

∣

∣

an+1

an

∣

∣

∣

∣

)

where L, U ∈ R.

(a) If L > 1, the series ∑∞n=1 an converges absolutely.

(b) If U < 1, ∑∞n=1 |an| = ∞ but the series ∑∞


(c) If L = U = 1, the series ∑∞n=1 |an| can converge or diverge, and similarly for ∑∞

n=1 an.


3.16 Theorem GAUSS CRITERION FOR CONVERGENCE OF A SERIES. Let {an} ⊆ C, {cn} ⊆ R andsuppose that

∣

∣

∣

∣

an+1

an

∣

∣

∣

∣

= 1− L

n+

cn

np

where p > 1 and |cn| ≤ M < ∞ , ∀n.

(a) If L > 1, the series ∑∞n=1 an converges absolutely.

(b) If L ≤ 1, ∑∞n=1 |an| = ∞ but ∑∞


3.17 Theorem INTEGRAL CRITERION FOR CONVERGENCE OF A SERIES. Let f (x), x ∈ R, a real-valuedcontinuous function, non-negative and non decreasing for x ≥ A, and let {an} ⊆ C a sequence such that|an| = f (n) for n ≥ A. Then

(a)∫ ∞

A f (x)dx < ∞ ⇒ ∑∞n=1 an converges absolutely;

(b)∫ ∞

A f (x)dx = ∞ ⇒ ∑∞n=1 |an| = ∞ .


3.18 Theorem DIRICHLET CRITERION FOR CONVERGENCE OF A SERIES OF PRODUCTS. Let {an} ⊆ C

and {bn} ⊆ R two sequences such that

(a)∣

∣∑Nn=1 an

∣

∣≤ M, for all N ≥ 1, where M ≥ 0 ,

(b) bn+1 ≤ bn, ∀n ,

(c) limn→∞

bn = 0 .

Then the series ∑∞n=1 anbn converges.

3.19 Corollary ALTERNATING SERIES CONVERGENCE CRITERION (LEIBNIZ). Let {an} ⊆ R a sequencesuch that

(a) |an+1| ≤ |an|, ∀n ,

(b) an = (−1)n+1 |an|, ∀n ,

(c) limn→∞

an = 0 .

Then the series ∑∞n=1 an converges and ∑∞

n=1 an ≤ a1 .


3.20 Theorem ABEL CRITERION FOR CONVERGENCE OF A SERIES OF PRODUCTS. Let {an} ⊆ C and{bn} ⊆ R be two sequences such that

(a) ∑∞n=1 an converges ,

(b) bn is a monotonic bounded sequence.

Then the series ∑∞n=1 anbn converges.

3.21 Remark In contrast with most criteria described above, the Abel and Dirichlet criteria do not entailabsolute convergence.

3.22 Theorem ABEL-DINI CRITERION FOR CONVERGENCE OF A SERIES OF RATIOS. If the series∑∞

n=1 an diverges such that SN = ∑Nn=1 an > 0, ∀N, and SN −→

N→∞∞, then

(a) the series ∑∞n=1 an/Sδ

n diverges for any δ ≤ 1,

(b) the series ∑∞n=1 an/Sδ

n converges for any δ > 1 .

3.23 Theorem LANDAU CRITERION FOR CONVERGENCE OF A SERIES OF PRODUCTS. Let {an} ⊆ R.The series ∑∞

n=1 |an|p , where p > 1, converges ⇔ the series ∑∞n=1 anbn converges for all sequences {bn} ⊆ C

such that ∑∞n=1 |bn|q converges, where q = p/(p−1) .

3.24 Remark The Landau theorem implies: if ∑∞n=1 |an|p < ∞ and ∑∞

n=1 |bn|q < ∞, where p > 1 and q =

p/(p−1) , then the series ∑∞n=1 anbn and ∑∞

n=1 |anbn| convergent. Further, it gives a necessary condition forthe convergence of ∑∞

n=1 |an|p when p > 1.


3.25 Theorem CONVERGENCE OF AN ARITHMETICALLY WEIGHTED MEAN. Let {an} ⊆ C. If the series∑∞

n=1 an converges, then

limN→∞

N

∑n=1

(1− n

N)an =

∞

∑n=1

an

and

limN→∞

N

∑n=1

n

Nan = lim

N→∞

1N

N

∑n=1

n an = 0 .

3.26 Theorem CESARO CONVERGENCE. Let {an} ⊆ C a sequence such that an → a ∈ C. Then

limN→∞

1N

N

∑n=1

an = a .

4. CONVERGENCE OF TRANSFORMED SERIES 21

4. Convergence of transformed series

4.1 Theorem CONVERGENCE OF LINEARLY TRANSFORMED SERIES. Let {an}∞n=0 ⊆ C and {bn}∞

n=0 ⊆ C

be two sequences such that∞

∑n=0

an = A and∞

∑n=0

bn = B

where A, B ∈ C. Then the sequences ∑∞n=0(an +bn) and ∑∞

n=0 can converge for any c ∈ C, and∞

∑n=0

(an +bn) = A+B,∞

∑n=0

can = c .

4.2 Definition CONVOLUTION. Let {an}∞n=0 ⊆ C and {bn}∞

n=0 ⊆ C. We call the sequence

cn =n

∑k=0

akbn−k,n = 0,1,2, . . .

the convolution of the sequences {an}∞n=0 and {bn}∞

n=0. Further, the series ∑∞n=0 cn is called the product of

the series ∑∞n=0 an and ∑∞

n=0 bn.

4.3 Remark We denote the convolution of the sequences an and bn by an ∗bn:

an ∗bn =n

∑k=0

akbn−k .


4.4 Theorem SUFFICIENT CONDITION FOR CONVERGENCE OF THE PRODUCT OF TWO SERIES

(CAUCHY-MERTENS). . Let {an}∞n=0 ⊆ C and {bn}∞

n=0 ⊆ C be two sequences such that

(a) ∑∞n=0 an = A and ∑∞

n=0 bn = B, where A, B ∈ C , and

(b) ∑∞n=0 an converges absolutely.

Then the series ∑∞n=0 cn , where cn = ∑n

k=0 akbn−k, converges and∞

∑n=0

cn = AB . (Mertens)

If, furthermore, ∑∞n=0 bn converges absolutely, the series ∑∞

n=0 cn converges absolutely.

4.5 Theorem LIMIT OF THE PRODUCT OF TWO SERIES (ABEL). If {an}∞n=0 ⊆ C, {bn}∞

n=0 ⊆ C and{cn}∞

n=0 ⊆ C are three sequences such that the series ∑∞n=0 an, ∑∞

n=0 bn and ∑∞n=0 cn converge to A, B and C

respectively, and if

cn =n

∑k=0

akbn−k ,

thenC = AB .

4.6 Theorem NECESSARY AND SUFFICIENT CONDITION FOR CONVERGENCE OF THE PRODUCT OF TWO

SERIES. Let {an} ⊆ R. The series ∑∞n=0 cn , where cn = ∑∞

k=0 akbn−k , converges for all sequences {bn} ⊆ R

such that ∑∞n=0 bn converges ⇔ ∑∞

n=0 |an| < ∞.


4.7 Theorem AGGREGATION OF TERMS IN A SERIES. Let {an}∞n=0 ⊆C be a sequence such that ∑∞

n=0 an =

A ∈ C, {rn}∞n=0 ⊆ N0 a monotonically increasing sequence of nonnegative integers such that r0 = 0 and

rn → ∞, and {bn}∞n=1 the sequence defined by

bn =rn−1

∑k=rn−1

ak,n = 1,2, . . . .

Then the series ∑∞n=1 bn converges and

∞

∑n=1

bn = A .

4.8 Definition REARRANGEMENT. Let {kn}∞n=0 be a sequence of nonnegative integers such that each

nonnegative integer appears once and only once in the sequence. If

a′n = akn, n = 0,1,2, . . . ,

we call the series ∑∞n=0 a′

n a rearrangement of the series ∑∞n=0 an.

4.9 Definition COMMUTATIVELY CONVERGENT SERIES. Let {an}∞n=0 ⊆C be a sequence such that ∑∞

n=0 an

converges to A ∈ C. The series ∑∞n=0 an is commutatively convergent if all the rearrangements ∑∞

n=0 a′n of the

series ∑∞n=0 an converge to A.


4.10 Theorem REARRANGEMENT OF AN ABSOLUTELY CONVERGENT SERIES (DIRICHLET). Let{an}∞

n=0 be a sequence such that ∑∞n=0 an converges absolutely to A ∈ C. Then all the rearrangements of

the series ∑∞n=0 an converge to A.

4.11 Theorem REARRANGEMENT OF A CONDITIONNALLY CONVERGENT SERIES (RIEMANN). Let{an}∞

n=0 ⊆ R be a series such that the series ∑∞n=0 an converges conditionally and let

−∞ ≤ α ≤ β ≤ ∞ .

Then there is a rearrangement ∑∞n=0 a′

n such that

liminfn→∞

(

n

∑m=0

a′m

)

= α , limsupn→∞

(

n

∑m=0

a′m

)

= β .

4.12 Theorem EQUIVALENCE BETWEEN ABSOLUTE AND COMMUTATIVE CONVERGENCE. Let {an} ⊆C be a sequence such that the series ∑∞

n=0 an converges. Then ∑∞n=0 an converges absolutely iff ∑∞

n=0 an

converges commutatively.


4.13 Theorem CONDITION FOR DOUBLE SERIES COMMUTATIVITY. Let {amn : m, n = 0, 1, 2, ...} ⊆ C

be a double sequence such that∞

∑n=0

|amn| = bm,m = 0,1,2, . . .

and ∑∞m=0 bm converges. Then

∞

∑m=0

∞

∑n=0

amn =∞

∑n=0

∞

∑m=0

amn.

5. UNIFORM CONVERGENCE 26

5. Uniform convergence

5.1 Notation In this section, fn and f refer to functions on a set E to the complex numbers C, i.e. fn : E →C

and f : E → C.

5.2 Definition UNIFORM CONVERGENCE. We say that the sequence of functions { fn}∞n=0 converges

uniformly on E to the function f if, for any ε > 0, there is an integer N such that

n ≥ N ⇒ | fn(x)− f (x)| < ε,∀x ∈ E .

In this case, we write fn → f uniformly on E.

5.3 Remark We dit that the series ∑∞i=0 fi(x) converges uniformly on E if the sequence of partial sums

sn(x) = ∑ni=0 fi(x), n = 0, 1 , . . . converges uniformly on E.


5.4 Theorem CAUCHY CRITERION FOR UNIFORM CONVERGENCE. The sequence of functions { fn}∞n=0

converges uniformly on E to a function f if and only if, for any ε > 0, there is an integer N such that

m,n ≥ N ⇒ | fm(x)− fn(x)| < ε,∀x ∈ E .

5.5 Theorem SUPREMUM CRITERION FOR UNIFORM CONVERGENCE. Suppose

limn→∞

fn(x) = f (x),∀x ∈ E ,

and letMn = sup

x∈E

| fn(x)− f (x)| .

Then, fn → f uniformly on E if and only if limn→∞

Mn = 0.

5.6 Theorem WEIERSTRASS UNIFORM CONVERGENCE CRITERION. Suppose | fn (x)| ≤ Mn , ∀x ∈ E ,n = 0,1, 2, ... and ∑∞

n=0 Mn < ∞ . Then the series ∑∞n=0 fn (x) converges uniformly on E.

5.7 Theorem UNIFORM CONVERGENCE AND CONTINUITY. If { fn}∞n=0 is a sequence of continuous

functions on E and if fn → f uniformly on E, then the function f is continuous on E.

5.8 Remark A sequence of continuous functions { fn}∞n=0 can converge to a continuous function f without

uniform convergence.


5.9 Theorem CONDITIONS OF UNIFORM CONVERGENCE (DINI). If

(a) K is a compact set,

(b) { fn}∞n=0 is a sequence of continuous functions on K,

(c) limn→∞

fn (x) = f (x), ∀x ∈ K, where f is a continuous function on K,

(d) fn (x) ≥ fn+1 (x), ∀x ∈ K, n = 0, 1, 2, .... ,

then fn → f uniformly on K.

5.10 Theorem UNIFORM CONVERGENCE AND DIFFERENTIATION OF FUNCTIONS OF REAL VARIABLES.Suppose [a, b]⊆ E ⊆ R and let fn : E → C, a sequence of differentiable functions on the interval [a,b] such

that the sequence { fn(x0)}∞n=0 converges for at least one x0 ∈ [a,b]. If the sequence { f ′n}∞

n=0 convergesuniformly on [a,b], then { fn}∞

n=0 converges uniformly on [a,b] to a differentiable function f on E, and

f ′ (x) = limn→∞

f ′n (x) ,∀x ∈ [a,b] .

5.11 Theorem UNIFORM CONVERGENCE AND DIFFERENTIATION OF FUNCTIONS OF COMPLEX VARI-ABLES. Let E ⊆ C and fn : E → C, n = 0, 1, 2, ... , a sequence of differentiable functions on E. If thesequence { fn}∞

n=0 converges to a function f in E and { f ′n}∞n=0 converges uniformly in E, then the function f

is differentiable in E andf ′(z) = lim

n→∞f ′n(z),∀z ∈ E.

6. PROOFS AND REFERENCES 29

6. Proofs and references

1. Rudin (1976, Chapters 1 and 3).1.15. Royden (1968, Section 2.4, pp. 35-38).1.22 – 1.23 Iyanaga and Kawada (1977, article 374, p. 1162).2. Rudin (1976, Chapter 3).2.2. Ahlfors (1979, Chapter 1, p. 62), Gillert, Küstner, Kellwich and Kästner (1986, Section 18.1, p.

422), and Rudin (1976, Chapter 3).2.6. Rudin (1976, Theorem 3.20, p. 57), and Gillert et al. (1986, Section 18.1, p. 420).3.1. Rudin (1976, Chapter 3).3.2. Gillert et al. (1986, Section 18.2, p. 428).3.3. Knopp (1956, Section 2.6.2, Theorem 1, p. 49, and Section 3.3, Theorem 1, p. 61).3.15. Devinatz (1968, section 3.2.4, p. 112) and Taylor (1955, Section 17.4, pp. 567-568).3.16. Taylor (1955, Section 17.4, pp. 568-569).3.17. Taylor (1955, Section 17.21, pp. 551-553).3.18. Anderson (1971, Lemma 8.3.1, p. 460).3.19. Piskounov (1980, 1980, chapitre XVI(7), pp. 294-296).3.22. Hardy, Littlewood and Polya (1952, Theorem 162, p. 120) and Knopp (1956, Section 2.6.2,

Theorem 1, p. 49, and Section 3.3, Theorem 1, p. 61).3.23. Beckenbach and Bellman (1965, Chapter 3, Theorem 11, pp. 116-117).3.25. Fuller (1976, Lemma 3.1.5, p. 112).4. Rudin (1976, Chapter 3).

6. PROOFS AND REFERENCES 30

4.4. Rudin (1976, Section 3.50, pp. 74-75), Taylor (1955, Section 17.6, pp. 575-580) and Iyanaga andKawada (1977, article 374, p. 1162).

4.6. Beckenbach and Bellman (1965, Chapter 3, Theorem 12, p. 117).4.7. Gillert et al. (1986, Chapter 18, p. 428).4.9. Gillert et al. (1986, Section 18.2, p. 429).4.10 - 4.11. Iyanaga and Kawada (1977, article 374, p. 1162).4.12. Gillert et al. (1986, section 18.2, p. 429).5. Ahlfors (1979, Chapter 2) and Rudin (1976, Chapter 7).5.2. Ahlfors (1979, Section 2.3, p. 36) and Rudin (1976, Definition 7.7, p. 147).5.4. Ahlfors (1979, Section 2.3, p. 36) and Rudin (1976, Theorem 7.8, p. 147).5.5. Rudin (1976, Theorem 7.9, p. 148).5.6. Rudin (1976, Theorem 7.10, p. 148).5.7. Ahlfors (1979, Section 2.3, p. 36) and Rudin (1976, Theorem 7.12, p. 150).5.8. Rudin (1976, Sections 7.12 and 7.6, p. 150 and 146).5.9. Rudin (1976, Theorem 7.13, p. 150).5.10. Rudin (1976, Theorem 7.17, p. 152).

Other useful references include: Devinatz (1968), Gradshteyn and Ryzhik (1980), Rudin (1987), andSpiegel (1964).

REFERENCES 31

References

Ahlfors, L. V. (1979), Complex Analysis: An Introduction to the Theory of Analytic Functions of One

Complex Variable, International Series in Pure and Applied Mathematics, third edn, McGraw-Hill,New York.

Anderson, T. W. (1971), The Statistical Analysis of Time Series, John Wiley & Sons, New York.

Beckenbach, E. F. and Bellman, R. (1965), Inequalities, second edn, Springer-Verlag, Berlin and New York.

Devinatz, A. (1968), Advanced Calculus, Holt, Rinehart and Winston, New York.

Fuller, W. A. (1976), Introduction to Statistical Time Series, John Wiley & Sons, New York.

Gillert, W., Küstner, Kellwich, H. and Kästner, H. (1986), Petite encyclopédie des mathématiques, ÉditionsK. Pagoulatos, Paris and Athens.

Gradshteyn, I. S. and Ryzhik, I. M. (1980), Table of Integrals, Series, and Products. Corrected and Enlarged

Edition, Academic Press, New York.

Hardy, G., Littlewood, J. E. and Polya, G. (1952), Inequalities, second edn, Cambridge University Press,Cambridge, U.K.

Iyanaga, S. and Kawada, Y., eds (1977), Encyclopedic Dictionary of Mathematics, Volumes I and II, 1977,Cambridge, Massachusetts.

Knopp, K. (1956), Infinite Sequences and Series, Dover Publications, New York.

REFERENCES 32

Piskounov, N. (1980), Calcul différentiel et intégral, Éditions de Moscou, Moscow.

Royden, H. L. (1968), Real Analysis, second edn, MacMillan, New York.

Rudin, W. (1976), Principles of Mathematical Analysis, Third Edition, McGraw-Hill, New York.

Rudin, W. (1987), Real and Complex Analysis, third edn, McGraw-Hill, New York.

Spiegel, M. R. (1964), Theory and Problems of Complex Variables with an Introduction to Conformal

Mapping and its Applications, Schaum Publishing Company, New York.

Taylor, A. E. (1955), Advanced Calculus, Blaisdell Publishing Company, Waltham, Massachusetts.

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