# Analytical solution for an almost geometric series

Is there any way of solving explicitly the limit of the series

$\sum_{n=0}^\infty q^n a^{p ^ n}$

where $0<p,q<1$ and $a>0$?

The series is obviously convergent as $a^{p ^ n} < \max(a,1)$ and so you can bound it by a standard geometric series, but it would be helpful to have an exact expression for it as well.

Thanks very much!

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Eep, a nested exponential. I doubt this has a tidy closed form... –  Guess who it is. Jan 26 '12 at 16:17
Tidy or not, I'm pretty sure there is no closed form except in the trivial case $a=1$. –  Robert Israel Jan 27 '12 at 0:12