Sharp bounds for the Bernoulli numbers
type: Item
instance of: scholarly article
title: Sharp bounds for the Bernoulli numbers
zbMATH Open document ID: 0960.11016
author: Horst Alzer
published in: Archiv der Mathematik
publication date: 2001-05-10T00:00:00Z
review text: The classical Bernoulli numbers \(B_n\) \((n=0,1,2\ldots)\) can be defined by \[ \frac{x}{e^x-1}=\sum_{n=0}^\infty B_n\frac{x^n}{n!}, \] \(|x|< 2\pi\). In the paper under review the author gives the best possible constants \(\alpha\) and \(\beta\) such that \[ \frac{2(2n)!}{(2\pi)^{2n}}\frac 1{1-2^{\alpha-2n}} \leq |B_{2n}|\leq \frac{2(2n)!}{(2\pi)^{2n}}\frac 1{1-2^{\beta-2n}} \] holds for all integers \(n\geq 1\), namely \(\alpha=0\) and \(\beta=2+\frac{\log(1-6/\pi^2)}{\log(2)}\approx 0.6491\ldots\).
reviewed by: Helmut Müller
Mathematics Subject Classification ID: 11B68
zbMATH DE Number: 1449201
zbMATH Keywords: Bernoulli numbers
MaRDI profile type: Publication
full work available at URL: s000130050432
OpenAlex ID: W1984938431
DOI: 10.1007/S000130050432
description: scientific article; zbMATH DE number 1449201