SumP[k, m] is 2k−1∑i=12m(1+(−1)i)/ik2^{k-1}\sum _{i=1}^{2m}\left(1+(-1)^i\right)/i^k2k−1∑i=12m(1+(−1)i)/ik.
SumP[k, m]
Overview, SumS, SumT.
SumP[1, m - 1]
S1′(m−1)S_1^{'(m-1)}S1′(m−1)
SumP[2, m - 1]
S2′(m−1)S_2^{'(m-1)}S2′(m−1)
SumP[1, m]
S1′(m)S_1^{'(m)}S1′(m)
SumP[1, 4]
2512\frac{25}{12}1225
Explicit[SumP[1, n/2]] % /. n -> 8
12(1−(−1)n)S1(n−12)+12((−1)n+1)S1(n2)\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)21(1−(−1)n)S1(2n−1)+21((−1)n+1)S1(2n)