Evaluation of Harmonic Sums with Integrals

Vivek Kaushik, Daniele Ritelli

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…