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Ramanujan Summation of Divergent Series

  • Bernard Candelpergher

Part of the Lecture Notes in Mathematics book series (LNM, volume 2185)

Table of contents

  1. Front Matter
    Pages i-xxiii
  2. Bernard Candelpergher
    Pages 1-29
  3. Bernard Candelpergher
    Pages 31-60
  4. Bernard Candelpergher
    Pages 61-111
  5. Bernard Candelpergher
    Pages 113-155
  6. Bernard Candelpergher
    Pages 157-173
  7. Back Matter
    Pages 175-195

About this book

Introduction

The aim of this monograph is to give a detailed exposition of the summation method that Ramanujan uses in Chapter VI of his second Notebook. This method, presented by Ramanujan as an application of the Euler-MacLaurin formula, is here extended using a difference equation in a space of analytic functions. This provides simple proofs of theorems on the summation of some divergent series. Several examples and applications are given. For numerical evaluation, a formula in terms of convergent series is provided by the use of Newton interpolation. The relation with other summation processes such as those of Borel and Euler is also studied. Finally, in the last chapter, a purely algebraic theory is developed that unifies all these summation processes. This monograph is aimed at graduate students and researchers who have a basic knowledge of analytic function theory.

Keywords

Ramanujan Divergent Series Summation Euler-MacLaurin formula Borel Summation Euler Summation

Authors and affiliations

  • Bernard Candelpergher
    • 1
  1. 1.Laboratoire J.A. Dieudonné. CNRSUniversité de Nice, Côte d’AzurNiceFrance

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-63630-6
  • Copyright Information Springer International Publishing AG 2017
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-63629-0
  • Online ISBN 978-3-319-63630-6
  • Series Print ISSN 0075-8434
  • Series Online ISSN 1617-9692
  • Buy this book on publisher's site