<s>[INST] Find $\sum_{n=0}^\infty\frac{2^n}{3^{2^{n-1}}+1}$ Find $\sum_{n=0}^\infty\frac{2^n}{3^{2^{n-1}}+1}$. 
I already proved that it converges but I can't find the sum, Although I think it should be $1+\frac{1}{\sqrt3 +1}$, from calculations.
 [/INST] Start with following infinite product expansion which is valid for $|q| < 1$,
$$\prod_{n=0}^\infty \left(1 + q^{2^n}\right) = \sum_{n=0}^\infty q^n = \frac{1}{1-q}$$
Taking logarithm and apply $q\frac{\partial}{\partial q}$ on both sides, one get
$$\sum_{n=0}^\infty \frac{2^n q^{2^n}}{1 + q^{2^n}} = \frac{q}{1-q}$$
Substitute $q$ by $\frac{1}{\sqrt{3}}$, one obtain
$$\sum_{n=0}^\infty \frac{2^n}{3^{2^{n-1}}+1} = \frac{1}{\sqrt{3}-1} = \frac{\sqrt{3}+1}{2} = 1 + \frac{\sqrt{3}-1}{2} = 1 + \frac{1}{\sqrt{3}+1}$$
the expression you expected.
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