Ramanujan Summation of Divergent Series
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 dif
- PDF / 2,123,704 Bytes
- 211 Pages / 439.43 x 666.14 pts Page_size
- 72 Downloads / 223 Views
Bernard Candelpergher
Ramanujan Summation of Divergent Series
Lecture Notes in Mathematics Editors-in-Chief: Jean-Michel Morel, Cachan Bernard Teissier, Paris Advisory Board: Michel Brion, Grenoble Camillo De Lellis, Zurich Alessio Figalli, Zurich Davar Khoshnevisan, Salt Lake City Ioannis Kontoyiannis, Athens Gábor Lugosi, Barcelona Mark Podolskij, Aarhus Sylvia Serfaty, New York Anna Wienhard, Heidelberg
2185
More information about this series at http://www.springer.com/series/304
Bernard Candelpergher
Ramanujan Summation of Divergent Series
123
Bernard Candelpergher Laboratoire J.A. Dieudonné. CNRS Université de Nice Côte d’Azur, Nice, France
ISSN 0075-8434 Lecture Notes in Mathematics ISBN 978-3-319-63629-0 DOI 10.1007/978-3-319-63630-6
ISSN 1617-9692 (electronic) ISBN 978-3-319-63630-6 (eBook)
Library of Congress Control Number: 2017948388 Mathematics Subject Classification (2010): 40D05, 40G05, 40G10, 40G99, 30B40, 30B50, 11M35, 11M06 © Springer International Publishing AG 2017 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
Introduction: The Summation of Series
The strange sums C1 X
n D 0 C 1 C 2 C 3 C 4 C 5 C 6 C 7 C 8 C 9 C :::
n0
and C1 X
n3 D 0 C 13 C 23 C 33 C 43 C 53 C 63 C 73 : : :
n0
appear in physics about the study of the Casimir effect which is the existence of an attractive force between two parallel conducting plates in the vacuum. These series are examples of divergent series in contrast to convergent series; the notion of convergence for a series was introduced by Cauchy in his Cours d’Analyse in order toPavoid frequent mistakes in working with series. Given a series of complex numbers n0 an , Cau
Data Loading...
