On the alternating series \(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots\)
type: Item
instance of: scholarly article
title: On the alternating series \(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots\)
zbMATH Open document ID: 1025.40001
DOI: 10.1016/S0022-247X(02)00344-X
author: József Bukor
author: László Tóth
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-07-30T00:00:00Z
review text: The remainder \(R_n:=\sum^\infty_{k=n+1} (-1)^{k+1} k^{-1}\) is provided best estimates \(a_n\leq|R_n|<b_n\), \(n\geq 1\). From what is known for alternating series in general, \(|R_n|< (n+1)^{-1}\). The first author had sharpened this by \(|R_n|< (2n+1)^{-1}\) and had given some estimate \((2n+\alpha)^{-1} <|R_n|\), in [Rev. Anal. Numér. Théor. Approx. 21, 195-199 (1992; Zbl 0801.40004)]. Now it is shown that, in fact, \((2n+1)^{-1} =b_n\) and that \(a_n=(2n+a)^{-1}\) with \(a=x_1\) from \((2n+x_n)^{-1}=|R_n|\), that is \(a=(1-\log 2)^{-1}-2\). For the proofs, \((x_n)\) is examined.
Mathematics Subject Classification ID: 40A05
zbMATH DE Number: 1956899
zbMATH Keywords: alternating harmonic series
zbMATH Keywords: best remainder estimates
reviewed by: Hans-Heinrich Körle
MaRDI profile type: Publication
cites work: Q4040797
cites work: Zur Approximation von e durch \((1+1/n)^ n\). (On the approximation of e by \((1+1/n)^ n)\)
cites work: Q4293492
full work available at URL: s0022-247x(02)00344-x
OpenAlex ID: W2021706773
description: scientific article; zbMATH DE number 1956899