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We present some simple proofs of the well-known expressions for \[  \zeta(2k) = \sum_{m=1}^\infty \frac{1}{m^{2k}},  \qquad  \beta(2k+1) = \sum_{m=0}^\infty \frac{(-1)^m}{(2m+1)^{2k+1}}, \] where $k =1,2,3,\dots$, in terms of the Bernoulli and Euler polynomials. The computationis done using only the defining properties of these polynomials and employingtelescoping series. The same method also yields integral formulas for$\zeta(2k+1)$ and $\beta(2k)$.  In addition, the method also applies to series of type \[  \sum_{m\in\mathbb{Z}} \frac{1}{(2m-\mu)^s},  \qquad  \sum_{m\in\mathbb{Z}} \frac{(-1)^m}{(2m+1-\mu)^s}, \] in this case usingApostol-Bernoulli and Apostol-Euler polynomials.

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