Evaluation of harmonic sums with integrals
Evaluation of harmonic sums with integrals
We consider the sums $S(k)=\sum _{n=0}^{\infty }\frac {(-1)^{nk}}{(2n+1)^k}$ and $\zeta (2k)=\sum _{n=1}^{\infty }\frac {1}{n^{2k}}$ with $k$ being a positive integer. We evaluate these sums with multiple integration, a modern technique. First, we start with three different double integrals that have been previously used in the literature to show $S(2)=\pi ^2/8,$ …