Rational Zeta Series - Expressions in The Form of P-series

Expressions in The Form of P-series

Adamchik and Srivastava give

\sum_{n=2}^\infty n^m \left = 1\, + \sum_{k=1}^m k!\; S(m+1,k+1) \zeta(k+1)

and

\sum_{n=2}^\infty (-1)^n n^m \left = -1\, +\, \frac {1-2^{m+1}}{m+1} B_{m+1} \,- \sum_{k=1}^m (-1)^k k!\; S(m+1,k+1) \zeta(k+1)

where are the Bernoulli numbers and are the Stirling numbers of the second kind.

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