I've come across this infinite series: $\sum\_{n=0}^\infty x^{n^\alpha}$, with $0<x<1$ and $\alpha > 0$.


Does this series have a name and/or is there a method for computing it (besides brute force, obviously)? Thanks!



Is there any numerical technique to sum x^(n^alpha), n=0,1,...?
For $\alpha > 1$, this sum will converge extremely rapidly, even if $x$ is fairly close to $1$ (you can't express in floating point numbers those numbers close enough to $1$ where this would converge slowly). The only case that is difficult, convergence wise, is when $\alpha$ is very close to 0.


In that case, your best bet are Levin's U-transform and related algorithms. Even better, just link in to [GSL](http://www.gnu.org/software/gsl/), which [already implements that](http://www.gnu.org/software/gsl/manual/html_node/Series-Acceleration-References.html).


