Find the value of $$\sum\limits_{n=1}^{\infty} \frac{2^{n}}{x^{2^n}+1}$$
I recently came across a question in which we had to find the value of the above question. The question seemed simple at first glance but the term $2^n$ in the numerator is posing me with a problem. I could neither reduce it to a telescopic series nor could I compare it with any standard expansion. I've run out of ideas. Would someone please help me to solve this problem?
Thanks for help.