Formula:DLMF:25.11:E7

$\displaystyle {\displaystyle \HurwitzZeta@{s}{a} = \frac{1}{a^s} + \frac{1}{(1+a)^s} \left( \frac{1}{2} + \frac{1+a}{s-1} \right) + \sum_{k=1}^n \binom{s+2k-2}{2k-1} \frac{\BernoulliB{2k}}{2k} \frac{1}{(1+a)^{s+2k-1}} - \binom{s+2n}{2n+1} \int_1^\infty \frac{\PeriodicBernoulliB{2n+1}@{x}}{(x+a)^{s+2n+1}} \diff{x} }$

Constraint(s)

${\displaystyle {\displaystyle s\neq 1}}$ &
${\displaystyle {\displaystyle a>0}}$ &
${\displaystyle {\displaystyle n=1,2,3,\dots }}$ &
$\displaystyle {\displaystyle \realpart{s} > -2n}$

Proof

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Take ${\displaystyle {\displaystyle N=1}}$ in

$\displaystyle {\displaystyle \HurwitzZeta@{s}{a} = \sum_{n=0}^N \frac{1}{(n+a)^s} + \frac{(N+a)^{1-s}}{s-1} - s \int_N^\infty \frac{x-\floor{x}}{(x+a)^{s+1}} \diff{x} }$

and integrate by parts.