与$\zeta(2k 1)$函数有关的两类快速收敛级数

New Rapidly Convergent Series Concerning $\zeta(2k 1)$

中文关键词:  Riemann zeta函数  快速收敛级数.

英文关键词:Riemann zeta function  rapidly convergent series.

中文摘要:

      本文给出了两类级数$$\sum_{n=1}^{\infty}\frac{(2n-1)!\zeta(2n)}{(2n 2k)!}\alpha^{2n},\ \ \sum_{n=1}^{\infty}\frac{(2n-1)!\zeta(2n)}{(2n 2k 1)!}\beta^{2n}$$的$\zeta(2k 1)$函数形式的求和公式,其中$k$为正整数,$\alpha$可取1, $\displaystyle \frac{1}{2}$, $\displaystyle \frac{1}{3}$, $\dis

英文摘要:

      Values of new series $$\sum_{n=1}^{\infty}\frac{(2n-1)!\zeta(2n)}{(2n 2k)!}\alpha^{2n},\ \ \sum_{n=1}^{\infty}\frac{(2n-1)!\zeta(2n)}{(2n 2k 1)!}\beta^{2n}$$ are given concerning $\zeta(2k 1)$, where $k$ is a positive integer, $\alpha$ can be taken as $1$, $1/2$, $1/3$, $2/3$, $1/4$, $3/4$, $1/6$, $5/6$ and $\beta$ can be taken as $1$, $1/2$. Some previous results are included as special cases in the present paper and new series converges more rapidly than those exsiting results for $\alpha = 1/3$, or $\alpha = 1/4$, or $\alpha = 1/6$.