Find the value of $\sum\limits_{n=1}^{\infty} \frac{2^{n}}{x^{2^n}+1}$

Question

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.

Answer 1

Hint: Multiply by $x^{2^n}-1$ in the numerator and denominator.

Answer 2

The series is telescopic: note that $$\frac{2^{n}}{x^{2^n}+1}=\dfrac{2^n}{x^{2^{n}}-1} - \dfrac{2^{n+1}}{x^{2^{n+1}}-1}.$$