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Divergent Series

G. H. Hardy
American Mathematical Society/Chelsea
Publication Date: 
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Allen Stenger
, on

This book is primarily about summability, that is, various methods to assign a useful value to a divergent series, usually by forming some kind of mean of the partial summands. It has only a little bit about asymptotic series, that is, divergent series for which it is possible to obtain a good approximation to the desired value by truncating the series at a well-chosen term. The present book is an AMS Chelsea 1991 reprint of the 1949 Oxford University Press edition. (There were several corrected reprints by Oxford, running through 1973, but this reprint seems to be the original 1949 edition.) This was Hardy’s last book, as he died at the end of 1947 after the book was written and while it was still working its way through the press; several younger colleagues saw it through the proofreading process.

The summability material is primarily about Tauberian theorems, an area that was developed in the early 1900s by Hardy and J. E. Littlewood. Tauberian theorems say that if a series is summable by a particular method and satisfies an additional hypothesis (usually that it is not too wiggly) then it in fact converges, or in some cases that it is summable by another method. These theorems are important in prime number theory and other fields.

There is also material here on summability for analytic continuation and for multiplication of series, the Euler-Maclaurin summation formula, and a good exposition of Wiener’s general Tauberian theorem (which, despite the name, belongs more to Fourier analysis than to summability).

The treatment is classical, and omits more modern material such as functional analysis methods and the algebraic view of Wiener’s general Tauberian theorem. The treatment is specific rather than general, and each summability method is handled separately without much attempt to put the methods in a general framework. The book is a reference more than a textbook, as there are no exercises. Very Good Features: several detailed indices, and extensive bibliographic notes at the end fo each chapter.

Although the material is still valuable, there are better books today. In particular the Tauberian theory in the present book has been largely superseded by Korevaar’s Tauberian Theory: A Century of Developments (Springer, 2004). A good recent survey of summability (not as in-depth as the present book) is Mursaleen’s 2014 Applied Summability Methods. A comprehensive book (that I have not seen) covering both classical and functional-analytic methods is Boos’s Classical and Modern Methods in Summability (Oxford, 2000).

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.


  • Introduction
  • Some historical examples
  • General theorems
  • Special methods of summation
  • Arithmetic means (1)
  • Arithmetic means (2)
  • Tauberian theorems for power series
  • The methods of Euler and Borel (1)
  • The methods of Euler and Borel (2)
  • Multiplication of series
  • Hausdorff means
  • Wiener's Tauberian theorems
  • The Euler-MacLaurin sum formula
  • Appendix I. On the evaluation of certain definite integrals by means of divergent series
  • Appendix II. The Fourier kernels of certain methods of summation
  • Appendix III. On Riemann and Abel summability
  • Appendix IV. On Lambert and Ingham summability
  • Appendix V. Two theorems of M. L. Cartwright
  • List of books
  • List of periodicals
  • List of authors
  • List of definitions
  • General index