Taylor expansions and summation of series
type: Item
instance of: scholarly article
title: Taylor expansions and summation of series
zbMATH Open document ID: 0834.40004
author: Bernard Candelpergher
published in: Expositiones Mathematicae
publication date: 1996-03-31T00:00:00Z
review text: The definition of \(\sum_{n\geq 0} a_n= \alpha_0\) in common sense, viz. Cauchy's, amounts to the existence of a null sequence \((\alpha_0, \alpha_1, \dots)\) such that \(\alpha_n- \alpha_{n+1}= a_n\), \(n\geq 0\). When termed \((\text{id--D}) \alpha= a\), this ``differential equation'' gives rise to an algebraic concept of summation, with \(a\), \(\alpha\) being in some appropriate \(\mathbb{C}\)-linear space where linear operators simulate differentiation (D) and summation (S), discrete or not, besides evaluation (V) that maps onto the most standard ``convergence elements''. Spaces thus furnished are called of type Taylor since their elements have Taylor-like ``algebraic expansions''. If, e.g., \(\sum_{n\geq 0} a_n\) has Abel limit \(\lambda\), that equation above with \(a: z\mapsto \sum_{n\geq 0} a_n z^n\) and \(\text{D} \sum_{n\geq 0} \alpha_n z^n:= \sum_{n\geq 0} \alpha_{n+1} z^n\) admits, within the space of holomorphic \(f(z)\), \(|z|<1\), for which \(f(1-0)\) exists, a unique solution \(\alpha: z\mapsto \sum_{n\geq 0} \alpha_n z^n\), rendering \(\lambda= \alpha (0)\). Here, at Abel's method, the Taylor space may be considered its convergence field, contrary to Cauchy convergence with a Taylor space of null sequences. In case of the Euler and the Borel means, D is a discrete resp. a proper differential operator, indicating the latter method to be a ``continuous'' counterpart of the former one. Similarly, through a shift D operating on a discrete resp. a continuous argument, convergence is related to a generalized limit as was introduced by Ramanujan (and makes Euler's constant sum the harmonic series). The new concept is also used to naturally conceive the notion of series products, notably the Cauchy product. The quotation from \textit{G. H. Hardy} is misleading, due to print setting on pp. 98, 99.
Mathematics Subject Classification ID: 40G10
Mathematics Subject Classification ID: 47B37
zbMATH DE Number: 797133
zbMATH Keywords: Taylor expansions
zbMATH Keywords: Euler summation
zbMATH Keywords: Abel summation
zbMATH Keywords: Borel summation
zbMATH Keywords: algebraic concept in summability
zbMATH Keywords: generating functions
zbMATH Keywords: linear operators
zbMATH Keywords: Taylor space
zbMATH Keywords: Cauchy convergence
zbMATH Keywords: differential operator
zbMATH Keywords: generalized limit
zbMATH Keywords: harmonic series
zbMATH Keywords: series products
zbMATH Keywords: Cauchy product
reviewed by: Hans-Heinrich Körle
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 797133