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

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

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