Evaluate $\sum\limits_{n=1}^{\infty} \frac{2^n}{1+2^{2^n}}$

Question

How to evaluate the infinite series: $$\sum\limits_{n=1}^{\infty} \frac{2^n}{1+2^{2^n}}$$

Answer 1

We have, $$\displaystyle \dfrac{2^n}{2^{2^n}-1} - \dfrac{2^{n+1}}{2^{2^{n+1}}-1} = \frac{2^n}{2^{2^n}+1}$$

The sum telescopes !