  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 with multiple integration, a modern technique.
First, we start with three different double integrals that have been previously
used in the literature to show $S(2)=\pi^2/8,$ which implies Euler's identity
$\zeta(2)=\pi^2/6.$ Then, we generalize each integral in order to find the
considered sums. The $k$ dimensional analogue of the first integral is the
density function of the quotient of $k$ independent, nonnegative Cauchy random
variables. In seeking this function, we encounter a special logarithmic
integral that we can directly relate to $S(k).$ The $k$ dimensional analogue of
the second integral, upon a change of variables, is the volume of a convex
polytope, which can be expressed as a probability involving certain pairwise
sums of $k$ independent uniform random variables. We use combinatorial
arguments to find the volume, which in turn gives new closed formulas for
$S(k)$ and $\zeta(2k).$ The $k$ dimensional analogue of the last integral, upon
another change of variables, is an integral of the joint density function of
$k$ Cauchy random variables over a hyperbolic polytope. This integral can be
expressed as a probability involving certain pairwise products of these random
variables, and it is equal to the probability from the second generalization.
Thus, we specifically highlight the similarities in the combinatorial arguments
between the second and third generalizations.
