Compute \(\sum_{m = 1}^\infty \sum_{n = 1}^\infty \frac{m^2 n}{3^m (n3^m + m3^n)}.\)

Mathematica couldn't come up with an answer so I'm not going to try either.

Where did you get this problem?

Probably from some online math website.

Numerically, this seems to converge (very quickly) to 9/32. This is such a simple result I suspect there is a way of finding it analytically. However, I have so far failed to do so!!