  In this paper we establish a new summation method by expanding
$\prod_{k}(1-\frac{z}{a_{k}})^{-1}$ with two approaches: the Taylor expansion
and the infinite partial fraction decomposition. Here we focus on the case when
$a_{k}$ is arithmetic sequence. By this summation we obtain many equalities
involve Hurwitz zeta function and Gammma function.
