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!
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! 


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 Utransform and related algorithms. Even better, just link in to GSL, which already implements that. 


The $k$th power is the generating function for the ways of expressing $n$ as a sum of $k$ $\alpha$th powers of positive integers. However, I don't think this helps much even for most integer values of $\alpha$. When $\alpha = 1$, this is a geometric series. When $\alpha = 2$, this is a theta function related to the Jacobi triple product formula $$\prod_{m=1}^\infty (1x^{2m})(1+x^{2m1}y^2)(1+x^{2m1}y^{2}) = \sum_{n=\infty}^\infty x^{n^2}y^{2n}$$ since $$ (\frac 12 \text{RHS}+\frac12) \bigg\_{y=1} = \sum\_{n=0}^\infty x^{n^2} .$$ You may be able to compute the sum when $\alpha=2$ more efficiently using one of the integral formulas or other properties for elliptic theta functions. Other than that, I don't know of special cases. This doesn't say anything about whether some rapid series acceleration technique exists. 

