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We recall a proof of Euler's identity $\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$ involving the evaluation of a double integral.We extend the method to find Hurwitz Zeta series of the form $S(k,a)=\sum_{n\in \mathbb{Z}} \frac{(-1)^{nk}}{(an+1)^k},$ where $a \in \mathbb{N} \setminus\lbrace 1 \rbrace$ and $k \in \mathbb{N}.$ In particular, we consider a general$k$-dimensional integral over $(0,1)^k$ that equals the series representation$S(k,a).$ Then we use an algebraic change of variables that diffeomorphicallymaps $(0,1)^k$ to a $k$-dimensional hyperbolic polytope. We interpret theintegral as a sum of two probabilities, and find explicit representations ofsuch probabilities with combinatorial techniques.

A Comment on the Sums $\sum_{n \in \mathbb{Z}} \frac{(-1)^{nk}}{(an+1)^k}$