The computation of $\zeta (2k)$, $\beta (2k+1)$ and beyond by using telescoping series

Óscar Ciaurri, Luis M. Navas, Francisco J. Ruiz +1

We present some simple proofs of the well-known expressions for \[ ζ(2k) = \sum_{m=1}^\infty \frac{1}{m^{2k}}, \qquad β(2k+1) = \sum_{m=0}^\infty \frac{(-1)^m}{(2m+1)^{2k+1}}, \] w…