TITLE: What is the value of Exponential Geometric Sum?
QUESTION [8 upvotes]: The sum in question is 
$$\sum_{k=0}^\infty x^{a^k}. $$
This comes from a more difficult summation, but if I can find the value of this one then the other one is solved. 
Does anyone know the exact value of this? Any help in that direction would be really appreciated.

The original problem is in this post: A tricky infinite sum-- solution found numerically, need proof
$X Y^{\alpha} + X^2 Y^{\alpha + \alpha^2} +X^3 Y^{\alpha + \alpha^2 + \alpha^3} + ...=\sum\limits_{j = 1}^{\infty}X^j \prod\limits_{k = 1}^{j}Y^{\alpha^k}$
where $0 < \alpha < 1, 0 < X < 1$ and $Y>0$

REPLY [2 votes]: There seems to be no general closed expression for this. The problem is known with the keyword "lacunary series", and there is an informative wikipedia-page about this.