| New Rapidly Convergent Series Concerning $\zeta(2k 1)$ |
| Received:October 01, 2009 Revised:November 20, 2010 |
| Key Words:
Riemann zeta function rapidly convergent series.
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| Fund Project:Supported by the National Natural Science Foundation of China (Grant No.10571095), Ningbo Natural Science Foundation (Grant No.2009A610078) and Research Fund of Ningbo University (Grant No.xkl09042). |
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| Abstract: |
| 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 $\alpha = 1/6$. |
| Citation: |
| DOI:10.3770/j.issn:1000-341X.2011.03.018 |
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