Closed form of $\sum_{i=0}^{\infty}x^{a^i}$

Question

In my case, the sum is actually $$\sum_{i=0}^{\infty}(1-a)^{(2a)^i}$$ I am confused because I find anything about this type of series online, and it seems like there ought to be a closed form of some kind. I noted that that each term in the exponent is the $i$th term of a geometric series, but I couldn't find any simplification from there.

For the record this is not homework. It's just a random little puzzle I found online.

Answer 1

There is no closed form for these. However, the series $\sum_{i=0}^\infty z^{a^i}$ for positive integer $a$ are studied: you might look up lacunary function.