If $\sum_{n=1}^\infty \frac{a_n}{n^s} $ converges, does $\sum_{n=1}^\infty \frac{a_n}{(n+1)^s} $ also converge?
Yes, because $(n+1)^{s} = n^{s} + sn^{s1} + O(s^2n^{\sigma  2})$. The first series necessarily converges in the open halfplane strictly to the right of s, and converges absolutely in the halfplane strictly to the right of s + 1. I hope I am not doing homework from a course in analytic number theory here. 

