On rational approximation of \((1-x)^ \sigma\)
type: Item
instance of: scholarly article
title: On rational approximation of \((1-x)^ \sigma\)
zbMATH Open document ID: 0861.41008
DOI: 10.1007/BF01268927
author: Horst Alzer
published in: Archiv der Mathematik
publication date: 1996-10-27T00:00:00Z
review text: Let \(S_n(x)\) be the \(n\)th partial sum of the power series for \((1-x)^0\) given by: \(S_n(x)= \sum^n_{s=0} (\sigma+j_j-1)x^j\), hold for all \(x\in(0,1)\). Let \(n\geq 1\) and \(0\leq \sigma<1\). The author proves that \[ {{x^\sigma} \over {S_n(x)}} [1-(1-x)^\sigma]< {1\over {S_n(x)}}- (1-x)^\sigma< {{x^\beta} \over {S_n(x)}} [1-(1-x)^\sigma] \] are valid for all \(x\in (0,1)\) if and only if \(\alpha= \infty\) and \(\beta\leq n\) (where \(x^\infty:=0\)).
reviewed by: C. M. Joshi
Mathematics Subject Classification ID: 41A20
zbMATH DE Number: 938584
MaRDI profile type: Publication
cites work: A remark on Reddy's paper on the rational approximation of \((1- x)^{1/2}\)
cites work: A note on rational approximation to \((1-x)^{\alpha}\)
cites work: A note on rational approximation to \((1-x)^{1/2}\)
cites work: A note on rational approximation to \((1-x)^{\sigma}\)
full work available at URL: https://doi.org/10.1007/bf01268927
OpenAlex ID: W2062977377
description: scientific article; zbMATH DE number 938584