  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 diffeomorphically
maps $(0,1)^k$ to a $k$-dimensional hyperbolic polytope. We interpret the
integral as a sum of two probabilities, and find explicit representations of
such probabilities with combinatorial techniques.
