A DOUBLE INEQUALITY FOR THE AP?RY CONSTANT

A remarkable result which led to Apery's proof of the irrationality of zeta(3) is given by the rapidly convergent series zeta(3) Let . 5 n (-1)k-1(k!)2 Rn = zeta(3) - n-ary sumation 2 k3(2k)! k=1denote the remainder of the series. In this paper, we obtain an asymptotic expansion of (-1)nRn . Based on the obtained result, we establish the upper and lower bounds of (-1)nRn. As an application of the obtained bounds, we give an approximate value of zeta(3).

Automated summary

This article presents an important result in Apery's proof of the irrationality of zeta(3) by introducing a rapidly convergent series. The asymptotic expansion of (-1)^nRn is obtained and the upper and lower bounds of (-1)^nRn are established based on the obtained result. Finally, an approximate value of zeta(3) is given using these bounds.