Radii of convergence of power series
type: Item
instance of: scholarly article
title: Radii of convergence of power series
zbMATH Open document ID: 0937.40003
author: S. James Taylor
author: Tibor Šalát
author: János T. Tóth
published in: Real Analysis Exchange
publication date: 2000-05-21T00:00:00Z
review text: The power series \(\sum _{n=0}^\infty a_n x^n\) is represented by the sequence \((a_n)_{n=0}^\infty \) in the Fréchet metric space of sequences with the metric \[ d ( (a_n)_{n=0}^\infty , (b_n)_{n=0}^\infty) = \sum _{n=0}^\infty 2^{-n} \frac{|a_n - b_n|} {(1+|a_n-b_n|)}. \] \(\sigma \)-porous properties of the set of sequences with positive radius of convergence are proved. Moreover, the authors study the Hausdorff dimension of the set of infinite subseries with different radius of convergence than a given series. (Subseries are identified with the numbers from \((0,1]\) such that the \(n\)th summand is contained in the subseries iff 1 is at the \(n\)th position of the dyadic expansion).
Mathematics Subject Classification ID: 40A30
Mathematics Subject Classification ID: 30B10
Mathematics Subject Classification ID: 28A05
Mathematics Subject Classification ID: 28A78
zbMATH DE Number: 1449516
zbMATH Keywords: power series
zbMATH Keywords: radius of convergence
zbMATH Keywords: \(\sigma \)-porous set
zbMATH Keywords: Hausdorff dimension
reviewed by: Josef Tkadlec
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1449516