I'm studying for the Putnam Exam and am a bit confused about how to go about solving this problem.
Sum the series
$$
\sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} \frac{m^2n}{3^m(n3^m + m3^n)}.
$$
I've tried "splitting" the expression to see if a geometric sum pops up but that didn't get me anywhere. I've also tried examining the first few terms of the series for the first few values of $m$ to see if an inductive pattern emerged but no luck there either.
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