For a cube of side length $a$ with 2 opposite sides held at the same potential $V$, the potential at the center of the cube can be expressed in series form as
And I am trying to show that this simplifies to
$\\ \Phi(a/2,a/2,a/2) = \frac{V}{3}$
such that the potential at the center of the cube is just the average of the potential over the surface of the cube.
Apparently, one must understand how to converge this double sum which does not appear to easily separate into either a product or sum of only $n$ and $m$ terms. How would one approach showing that sum indeed converges to $\frac{\pi^2}{48}$?