TITLE: Double sum trouble
QUESTION [13 upvotes]: Evaluate:
$$\sum_{j=1}^{\infty} \sum_{i=1}^{\infty} \frac{j^2i}{3^j(j3^i+i3^j)}$$
Honestly, I don't see where to start with this. I am sure that this is a trick question and I am missing something very obvious. I tried writing down a few terms for a fixed $j$ but I couldn't spot any pattern or some kind of easier series to handle. 
Any help is appreciated. Thanks!

REPLY [10 votes]: Hint: Expanding in terms of parial fractions: 
$$\frac{1}{3^j (j 3^i + i 3^j)}=\frac{1}{j 3^i 3^j}-\frac{i}{j 3^i (i 3^j + j 3^i)}\\
\implies \frac{j^2i}{3^j (j 3^i + i 3^j)}=\frac{j i}{3^i 3^j}-\frac{j i^2}{3^i (i 3^j + j 3^i)}.$$