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