Summation of divergent series from the nonstandard point of view
type: Item
instance of: scholarly article
title: Summation of divergent series from the nonstandard point of view
zbMATH Open document ID: 0879.03024
published in: Real Analysis Exchange
publication date: 1997-12-10T00:00:00Z
review text: The authors perform a development of techniques of computations using nonstandard analysis. The known formula \(1+\tau +\tau ^2+\dots =\frac {1}{1-\tau}\) valid for \(\tau\) s.t. \(|\tau |<1\) and similar formulas are adapted also for \(\tau\) being the shift operator (raising the indices of the series by 1; to the series \(a_0+a_1+a_2+\dots\) the series \(b_0+b_1+b_2+\dots \) is assigned, where \(b_i=a_{i+1}\)). In this framework they describe the Hutton and Euler transform of series used for the summation of some divergent series. They show the necessity of adaption of the understanding of \(1\) (as a suitable matrix near to the unit matrix) and exponentiation (also in a nonstandard manner). It is obvious that they use matrix calculus for matrices of size \(\eta \times \eta\) (\(\eta\) being infinitely large) and standard part (shadow) technique. The paper is written in a very nice manner. The authors bring the reader's attention to some problems connected with the nonstandard work, too. I think that the paper gives a lot of motivations for thinking and I heartily recommend it for reading.
reviewed by: Karel Čuda
Mathematics Subject Classification ID: 03H05
Mathematics Subject Classification ID: 03E70
Mathematics Subject Classification ID: 40A25
Mathematics Subject Classification ID: 40J05
zbMATH DE Number: 1095842
zbMATH Keywords: divergent series
zbMATH Keywords: summability
zbMATH Keywords: nonstandard analysis
zbMATH Keywords: Hutton transform
zbMATH Keywords: Euler transform
author: Vladimir Kanovei
author: M. Reeken
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1095842