Evaluate $\sum \frac{2^k}{3^{2^k}+1}$ Evaluate the sum $$\sum_{k=0}^{\infty} \left ( \frac{2^k}{3^{2^k}+1} \right )$$
I first tried to see weather the sum actually converges, by ratio test it does as $\lim_{n\to \infty}|\frac{a_{n+1}}{a_n}|<1$ where $a_n$ is the nth term. Now I am left with finding the actual sum. I thought to write it so that it can telescope, but can't find any suitable representations. Any hints for the evaluation part? Any Taylor polynomials etc?
Update: Wolfram says its $1/2$

$$\frac{2^k}{3^{2^k}+1}=\frac{2^k}{3^{2^k}-1}-\frac{2^{k+1}}{3^{2^{k+1}}-1}$$
So telescoping is the way to go, here.
