Rational Zeta Series - Half-integer Power Series

Half-integer Power Series

Similar series may be obtained by exploring the Hurwitz zeta function at half-integer values. Thus, for example, one has

\sum_{k=0}^\infty \frac {\zeta(k+n+2)-1}{2^k} {{n+k+1} \choose {n+1}}=\left(2^{n+2}-1\right)\zeta(n+2)-1

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