Theory and Application of Infinite Series

This is a comprehensive reference on infinite series. It comes at the subject from a function-theoretic point of view, that is: what functions can be represented by infinite series (particularly power series and Fourier series, but also some more specialized ones), and what can we infer about a function from its series? It starts from the beginning, and develops and defines the concept of convergence and includes a lot of material on convergence tests, but it is primarily about functions. There is also a moderate amount of information about transformation of series to increase the speed of convergence, and numerous examples of numerical calculation of function values from series.

The book was first published in German in 1921 and went through several revisions, and there were two English editions. The present volume is a 1990 Dover reprint of the 1951 English edition from Blackie & Son, which is based on the 1947 German edition. The book has aged well, and most of the material here is still valuable today. It omits coverage of some series that have become important today, such as the hypergeometric series. There are a few spots where the language has changed, mostly in the summability sections, but this causes little problem in following the narrative. There is no computer coverage or discussion of round-off error, and all the numerical examples assume hand calculation and are worked to about a dozen decimal places.

The most valuable part of the book is the examples and exercises. These are numerous and interesting, and appear at the end of each section and of each chapter. They nearly always ask for facts or proofs about particular series or functions, rather than to prove general theorems that were not covered in the body.

I don’t know of any modern books with the same coverage as this one. A near-contemporary is Bromwich’s 1926 An Introduction to the Theory of Infinite Series, that covers much the same topics and to the same level of detail. A good modern book is Bonar & Khoury’s 2006 undergraduate book Real Infinite Series, although that is much more limited and is concerned mostly with convergence tests and to some extent with identifying and evaluating series. Books on real analysis, complex analysis, and Fourier analysis often have good sections on the properties of the corresponding series. Summability tends to be studied in the context of Fourier series, but is also the subject of a whole book by G. H Hardy, the 1949 Divergent Series.

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.