Asymptotic expansion at $x=1^-$ for $\sum_{n=1}^{\infty} x^{a_n}$

Question

I feel that this general fact should be known. Suppose I have a strictly increasing sequence of integer positive numbers $\{a_n\}_{n \in \mathbb{N}}$. We want to investigate the behavior of $$ f(x)= \sum_{n=0}^{\infty} x^{a_n}.$$ Of course this is a power series with convergence radius $R=1$. I want to know something about the limit $\lim_{x \to 1^-} (1-x)^{\alpha} f(x)$, depending on $\alpha$. Are there Tauberian/Abelian theorems which can apply? This comes from a particular problem when $a_n=n^2$ where I think I can show that $$\lim_{x \to 1^-} \sqrt{1-x} \cdot f(x) =\int_0^{\infty} e^{-x^2} \,dx = \frac {\sqrt{\pi}}2,$$ by looking at it as an approximated Riemann sum. I am interested also at behaviors of other similar series like $f(x)= \sum_{p \; prime} \frac{x^p}p$ etc...