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If $\sum_{n=1}^\infty \frac{a_n}{n^s} $ converges, does $\sum_{n=1}^\infty \frac{a_n}{(n+1)^s} $ also converge?

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Yes, because $(n+1)^{-s} = n^{-s} + sn^{-s-1} + O(|s|^2n^{-\sigma - 2})$. The first series necessarily converges in the open half-plane strictly to the right of s, and converges absolutely in the half-plane strictly to the right of s + 1. I hope I am not doing homework from a course in analytic number theory here.

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