I'm studying the series $$\sum_{n=1}^{\infty}\frac{1}{(1+n)^{-z}}$$
If $z = x+iy$, what is the behaviour of the series for $-1<x<0 \ $?
2026-04-03 00:25:20.1775175920
Series $\sum_{n=1}^{\infty}\frac{1}{(1+n)^{-z}} \ $, $ z \in \mathbb{C}$
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Your series is the Riemann zeta function thinly disguised:
$$\sum_{n=1}^{+\infty}\frac{1}{(1+n)^{-z}} = \zeta(-z)-1.$$