TITLE: Closed form formula for the following sum
QUESTION [6 upvotes]: Does anyone know of a closed-form formula for the sum $\sum_{n = 1}^\infty x^{2^n-1}$? We can assume that $0<x<1$. 
Thanks!

REPLY [2 votes]: Further discussion of this topic is available here (except that topic refers to partial sums). However, I did give an estimate in this answer, which gives:
$$ -\frac{\mathrm{Ei}\left(\log (x)\right)}{x\log (2)} \le \sum_{i=0}^\infty x^{2^i - 1} \le -\frac{\mathrm{Ei}\left(\frac{\log (x)}{2}\right)}{x\log (2)}  $$
Where $\mathrm{Ei}(x)$ is the Exponential Integral.