Ramanujan summation of divergent series
type: Item
instance of: scholarly article
title: Ramanujan summation of divergent series
zbMATH Open document ID: 1386.40001
DOI: 10.1007/978-3-319-63630-6
author: Bernard Candelpergher
published in: Lecture Notes in Mathematics
publication date: 2017-08-02T00:00:00Z
review text: Sometimes it is desirable to associate values to divergent series, i.e., ``sum'' them symbolically by following certain rules. For example, a divergent series can be ``evaluated'' by using Abel summation, Borel summation, Cesàro summation, or Cauchy summation. The present book discusses the theory of Ramanujan summation, which we will denote here by \((R)\sum _{n\geq 1}f(n)\). The theory comes from the works of Srinivasa Ramanujan. The summation technique is based on the Euler-Maclaurin summation formula and uses the partial sums of a given series. In some cases, one has the formula \((R)\sum _{n\geq 1}f(n) =\sum _{n=1}^{\infty }\left(f(n)-\int _{n}^{n+1}f(x)\,dx \right)\). It is also shown that, for functions of moderate growth, \((R)\sum _{n\geq 1}f(n) =\mathop{\lim }\limits_{z\to 0+} \left(\sum _{n=1}^{\infty }f(n)e^{-nz} -\int _{1}^{\infty }f(x)e^{-zx}\, dx \right)\). Here are several examples: \((R)\sum _{n\geq 1}\frac{1}{n} =\gamma \), \((R)\sum _{n\geq 1}\frac{1}{n^{s} } =\zeta (s)-\frac{1}{s-1} \quad (\operatorname{Re}s>1)\), \((R)\sum _{n\geq 1}(n-1)^{k} =\zeta (-k)\). (\(\zeta (s)\) is Riemann's zeta function and \(\gamma \) is Euler's constant.) When \(k=1\), the last example reduces to the popular paradoxical evaluation \((R)(1+2+3+\dots)=-\frac{1}{12} \) . The book contains a number of interesting formulas and makes several important connections to classical analysis.
Mathematics Subject Classification ID: 40-02
Mathematics Subject Classification ID: 40D05
Mathematics Subject Classification ID: 40G05
Mathematics Subject Classification ID: 40G10
Mathematics Subject Classification ID: 30B40
Mathematics Subject Classification ID: 11M06
Mathematics Subject Classification ID: 11M35
zbMATH DE Number: 6755646
zbMATH Keywords: divergent series
zbMATH Keywords: Euler-Maclaurin formula
zbMATH Keywords: Cesàro summation
zbMATH Keywords: Cauchy summation
zbMATH Keywords: Plana formula
zbMATH Keywords: Euler constant
reviewed by: Khristo N. Boyadzhiev
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-319-63630-6
OpenAlex ID: W2754757130
description: scientific article; zbMATH DE number 6755646