SumP[k, m] is 2^{k-1}\sum _{i=1}^{2m}\left(1+(-1)^i\right)/i^k.
SumP[1, m - 1]S_1^{'(m-1)}
SumP[2, m - 1]S_2^{'(m-1)}
SumP[1, m]S_1^{'(m)}
SumP[1, 4]\frac{25}{12}
Explicit[SumP[1, n/2]]
% /. n -> 8\frac{1}{2} \left(1-(-1)^n\right) S_1\left(\frac{n-1}{2}\right)+\frac{1}{2} \left((-1)^n+1\right) S_1\left(\frac{n}{2}\right)
\frac{25}{12}