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Divergent Series
Godfrey H. Hardy

AMS Chelsea Publishing
1949; 396 pp; hardcover
Volume: 334
ISBN-10: 0-8218-2649-2
ISBN-13: 978-0-8218-2649-2
List Price: US$54
Member Price: US$48.60
Order Code: CHEL/334.H
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From the Preface by J. E. Littlewood: "All [Hardy's] books gave him some degree of pleasure, but this one, his last, was his favourite. When embarking on it he told me that he believed in its value (as well he might), and also that he looked forward to the task with enthusiasm. He had actually given lectures on the subject at intervals ever since his return to Cambridge in 1931, and he had at one time or another lectured on everything in the book except Chapter XIII [The Euler-MacLaurin sum formula] ... [I]n the early years of the century the subject [Divergent Series], while in no way mystical or unrigorous, was regarded as sensational, and about the present title, now colourless, there hung an aroma of paradox and audacity."


Review of original edition ...

"This is an inspiring textbook for students who know the theory of functions of real and complex variables and wish further knowledge of mathematical analysis. There are no problems displayed and labelled "problems," but one who follows all of the arguments and calculations of the text will find use for his ingenuity and pencil. The book deals with interesting and important problems and topics in many fields of mathematical analysis, to an extent very much greater than that indicated by the titles of the chapters. It is, of course, an indispensable handbook for those interested in divergent series. It assembles a considerable part of the theory of divergent series, which has previously existed only in periodical literature. Hardy has greatly simplified and improved many theories, theorems and proofs. In addition, numerous acknowledgements show that the book incorporates many previously unpublished results and improvements of old results, communicated to Hardy by his colleagues and by others interested in the book."

-- Mathematical Reviews

Table of Contents

  • 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
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