You are here

Infinite Sequences and Series

Konrad Knopp
Dover Publications
Publication Date: 
Number of Pages: 
BLL Rating: 

The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Allen Stenger
, on

This is a short and streamlined but still comprehensive introduction to the theory of infinite series. It starts with the same prerequisites as the study of infinite series in calculus, but goes far beyond those accounts, which usually end with convergence tests and the rudiments of power series. This book is that rare thing, a Dover original. Knopp wrote the book in German towards the end of his life, and Frederick Bagemihl translated the manuscript into English for publication. The original German version was never published.

The selection of topics and depth of coverage is done very skillfully, so that on the one hand it gives useful results in nearly every part of the theory but on the other hand only goes deep enough to give you a good start; every subject introduced has been investigated much more deeply than is shown here.

That said, the book is still fairly conventional. It does start with a whirlwind construction of the real and complex numbers, based on Dedekind cuts. It develops the theory of convergence of sequences first and then switches to series.

It includes a few unusual topics. These include some summability and the behavior of weighted averages of sequences; these are used for proving some of the more delicate results on convergence rather than for Fourier series. It also develops a moderate amount of complex function theory, based solely on power series with no integration or differentiation. This includes the elementary functions, and proofs than the reciprocal of a power series and the inverse function of a power series can be developed in power series. The only conspicuous omissions are series of functions (except power series), uniform convergence, and infinite products.

The book is very different from Knopp’s “big” series book, Theory and Application of Infinite Series. The latter book goes much deeper, and is both a textbook and a scholarly work, with many exercises and references to the original papers. The present book only references other texts and encyclopedia articles. Oddly, there are no exercises in the present book, so it is not a text in the modern American sense. It does have a large number of worked examples, but most of the work is extremely brief. I found the present book much easier to follow than the “big” book, I think because it is more focused and has few digressions.

Bottom line: probably too hard to be an introductory text today, not because of the exposition but because of the lack of exercises and the conciseness of the worked examples. It would work well for good upper-division undergraduates who have already been exposed to infinite series in calculus and are curious to know more.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.


Chapter 1. Introduction and Prerequisites
  1.1 Preliminary remarks concerning sequences and series
  1.2 Real and complex numbers
  1.3 Sets of numbers
  1.4 Functions of a real and of a complex variable
Chapter 2. Sequences and Series
  2.1 Arbitrary sequences. Null sequences
  2.2 Sequences and sets of numbers
  2.3 Convergence and divergence
  2.4 Cauchy's limit theorem and its generalizations
  2.5 The main tests for sequences
  2.6 Infinite series
Chapter 3. The Main Tests for Infinite Series. Operating with Convergent Series
  3.1 Series of positive terms: The first main test and the comparison tests of the first and second kind
  3.2 The radical test and the ratio test
  3.3 Series of positive, monotonically decreasing terms
  3.4 The second main test
  3.5 Absolute convergence
  3.6 Operating with convergent series
  3.7 Infinite products
Chapter 4. Power Series
  4.1 The circle of convergence
  4.2 The functions represented by power series
  4.3 Operating with power series. Expansion of composite functions
  4.4 The inversion of a power series
Chapter 5. Development of the Theory of Convergence
  5.1 The theorems of Abel, Dini, and Pringsheim
  5.2 Scales of convergence tests
  5.3 Abel's partial summation. Lemmas
  5.4 Special comparison tests of the second kind
  5.5 Abel's and Dirichlet's tests and their generalizations
  5.6 Series transformations
  5.7 Multiplication of series
Chapter 6. Expansion of the Elementary Functions
  6.1 List of the elementary functions
  6.2 The rational functions
  6.3 The exponential function and the circular functions
  6.4 The logarithmic function
  6.5 The general power and the binomial series
  6.6 The cyclometric functions
Chapter 7. Numerical and Closed Evaluation of Series
  7.1 Statement of the problem
  7.2 Numerical evaluations and estimations of remainders
  7.3 Closed evaluations
  Bibliography; Index