Find the sum $\sum_{n=1}^{\infty} \frac{3^n}{5^n-2^{2n}}$. Can someone help me with this sum $\sum_{n=1}^{\infty} \frac{3^n}{5^n-2^{2n}}$. I can't find $S_n$

You can write $$\frac{3^n}{5^n-2^n} = \frac{(3/5)^n}{1 - (2/5)^n} = \sum_{k=0}^\infty (3/5)^n (2/5)^{kn}$$
so your sum is
$$ \eqalign{ \sum_{n=1}^\infty \sum_{k=0}^\infty (3/5)^n (2/5)^{kn}
&= \sum_{k=0}^\infty \sum_{n=1}^\infty (3/5)^n (2/5)^{kn}\cr
&= \sum_{k=0}^\infty \frac{3 \cdot 2^k}{5^{1+k}-3 \cdot 2^k}}$$
But that's no closer to a "closed form" than the original.
