TITLE: Finding the value of $\sum\limits_{k=0}^{\infty}\frac{2^{k}}{2^{2^{k}}+1}$
QUESTION [24 upvotes]: Does this weighted sum of reciprocals of Fermat numbers,
$$
F=\sum_{k=0}^{\infty}\dfrac{2^{k}}{2^{2^{k}}+1}
$$
have a nice closed form? Wolfram says it's $1$.
Thanks.

REPLY [4 votes]: Use the identity
$$\frac{1}{t+1} = \frac{1}{t-1} - \frac{2}{t^2 -1}.$$