Residue of Pole $s=1$ of $\zeta$ function

Question

I have trouble to understand Why the residue of the riemann $\zeta$ function is 1. I can just find that One can see this because $\lim_{s\to 1} (s-1)\zeta(s)=1$. But I do not understand how to get the 1 by using the Series representation $\zeta(s)=\sum_{n=1}^\infty \frac{1}{n^s}$