I have the following math problem:
$ \sum^{\infty}_{j=1} \sum^{\infty}_{i=1} C/(i*j)^2 = 1 $
where C is the value of the constant that makes this function be equal to 1.
I am not being able to find how to prove this converge and prove how C goes to $ 36/pi^4 $
On
$\sum\limits_{i=1}^{\infty} \frac 1 {i^{2}}=\frac {\pi^{2}} 6$. The given equation is $C \sum\limits_{i=1}^{\infty} \sum\limits_{i=1}^{\infty} \frac 1 {i^{2}j^{2}}=1$. You can sum w.r.t. $i$ first and then w.r.t. $j$. When you sum w.r.t. $i$, $\frac 1{j^{2}}$ comes out of the summation so you get $C\frac {\pi^{2}} 6 \frac {\pi^{2}} 6=1$.
Use the identity $\sum_{k=1}^{\infty}\frac{1}{k^2}=\frac{\pi^2}{6}$ two times. You will get the answer.