Asymptotic results of the remainders in the series representations for the Apery constant

In this paper, we present certain asymptotic expansions and prove some inequalities for the Apery constant. The results are based on the series representations of the Apery constant. For example, based on Gosper's result zeta(3) = 1/4 Sigma(infinity)(k=1) 30k-11/(2k - 1)k(3) ((2k)(k))(2), we establish the following asymptotic expansion of the remainder R-n: R-n = 1/4 Sigma(infinity)(k=n+1) 30k-11/(2k - 1)k(3) ((2k)(k))(2) similar to pi/n(2)2(4n+2) {1 - 7/4n + 81/32n(2) - 489/128n(3) + 13787/2048 - ... } as n -> infinity. Moreover, we give a formula for determining the coefficients in expansion. Then we prove inequalities for the Apery constant.

Automated summary

In this paper, certain asymptotic expansions and inequalities for the Apery constant are presented. The results are based on the series representations of the Apery constant. For example, based on Gosper's result zeta(3) = 1/4 Sigma(infinity)(k=1) 30k-11/(2k - 1)k(3) ((2k)(k))(2), the following asymptotic expansion of the remainder R-n is established: R-n = 1/4 Sigma(infinity)(k=n+1) 30k-11/(2k - 1)k(3) ((2k)(k))(2) similar to pi/n(2)2(4n+2) {1 - 7/4n + 81/32n(2) - 489/128n(3) + 13787/2048 - ... }, as n -> infinity. Moreover, a formula for determining the coefficients in the expansion is given. Inequalities for the Apery constant are also proven.