  In this paper we provide a new series representation for the values of
Riemann zeta function at integer arguments, namely: $
\zeta(m)=\sum_{n=1}^{\infty}\frac{m(-1)^{n-1}\Gamma(1-\omega_{m}n)...\Gamma(1-\omega_{m}^{m-1}n)}{n!n^m}$,
where $n$ is an integer that lager than $1$ and $\omega$ is the $m$-th root of
unity. This series converges quite fast. It's derived by some technique of
infinite partial fraction decomposition. With this technique we also establish
other useful formulas related to gamma function.
