On some inequalities for the Bernoulli numbers
type: Item
instance of: scholarly article
title: On some inequalities for the Bernoulli numbers
zbMATH Open document ID: 0830.11008
DOI: 10.1007/BF02844246
author: Ciro D'Aniello
publication date: 1996-02-01T00:00:00Z
review text: The well-known Fourier representation of the Bernoulli polynomial \(B_{2n} (x)\) gives \[ B_{2n}= B_{2n} (0)= {{(-1)^{n-1} 2(2n)!} \over {(2\pi )^{2n}}} \zeta (2n). \] Depending on the number of summands which were used to compute \(\zeta (2n)\) one gets different lower bounds. In the same manner the author obtains the lower bound in \[ {{2(2n)!} \over {(2\pi )^{2n}}} {1\over {1- 2^{-2n}}} <|B_{2n} |< {{2(2n)!} \over {(2\pi )^{2n}}} {1\over {1- 2^{1- 2n}}}, \] for \(n\in \mathbb{N}\), where the upper bound is well-known. To demonstrate the sharpness one checks that for \(n=10\) these inequalities mean \[ 529.1242423 <|B_{20} |< 529.1247469. \] A similar analysis can be done for the Euler numbers \(E_{2n}\).
reviewed by: Helmut Müller
Mathematics Subject Classification ID: 11B68
zbMATH DE Number: 790778
zbMATH Keywords: Bernoulli polynomial
zbMATH Keywords: lower bound
zbMATH Keywords: inequalities
zbMATH Keywords: Euler numbers
MaRDI profile type: Publication
cites work: Q5558293
cites work: Q3963081
cites work: The real zeros of the Bernoulli polynomials
published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 790778