On some inequalities for the Bernoulli numbers
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 \(E_{2n}\).
scientific article; zbMATH DE number 790778