Inequalities and asymptotics for the Euler-Mascheroni constant based on DeTemple's result
type: Item
instance of: scholarly article
title: Inequalities and asymptotics for the Euler-Mascheroni constant based on DeTemple's result
zbMATH Open document ID: 1360.41015
author: Chao-Ping Chen
published in: Numerical Algorithms
publication date: 2016-11-18T00:00:00Z
review text: The author considers the approximation of the Euler-Mascheroni constant \(\gamma\) as the limit of the difference between the harmonic series and the logarithm. In particular, DeTemple's inequalities: \[ {1\over24(n+1)^2}<\left[\sum_{k=1}^n{1\over k}-\ln\left(n+{1\over 2}\right)\right]-\gamma<{1\over24n^2},\quad n=1,2,3,\dots \] The author goes one step forward and shows that \[ \left[\sum_{k=1}^n{1\over k}-\ln\left(n+{1\over 2}\right)\right]-\gamma\sim\sum_{k=0}^\infty{a_k\over(n^2+n+b_k)^{2k-1}}, \] where the constants \(a_k\) and \(b_k\) are determined by means of certain nonlinear recurrence relations. Previously, the author has obtained a similar expansion for the digamma function. As a consequence, the following improvement of DeTemple's inequalities are given: \[ {1\over24(n^2+n+\alpha)}<\left[\sum_{k=1}^n{1\over k}-\ln\left(n+{1\over 2}\right)\right]-\gamma<{1\over24(n^2+n+\beta)}, \quad n=1,2,3,\dots \] for certain constants \(\alpha\) and \(\beta\). Another more elaborated couple of inequalities are also given.
Mathematics Subject Classification ID: 41A60
Mathematics Subject Classification ID: 11Y60
Mathematics Subject Classification ID: 33B15
zbMATH DE Number: 6654601
zbMATH Keywords: psi function
zbMATH Keywords: Euler-Mascheroni constant
zbMATH Keywords: asymptotic formula
zbMATH Keywords: inequality
reviewed by: José L. López
MaRDI profile type: Publication
full work available at URL: s11075-016-0116-9
OpenAlex ID: W2291518383
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DOI: 10.1007/S11075-016-0116-9
describes a project that uses: Digital Library of Mathematical Functions
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description: scientific article; zbMATH DE number 6654601