I am looking for ways to simplify the sum $$\sum_{n=1}^\infty \frac{1}{n(n+a)^b}, \quad a\in\mathbb{R}^+, b\in\mathbb{N}.$$ The first thought I had approaching this was to use Hurwitz and/or Zeta function. How should I go about evaluating this sum? (Mathematica doesn't have any simplification, but I need at least approximative form in tearms of $a$ and $b$ for simulations).
2026-03-29 13:41:08.1774791668
Evaluating the sum $\sum_{n=1}^\infty \frac{1}{n(n+a)^b}$
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