Karamata's Tauberian theorem states the following. Let $A(z)=\sum a_nz^n$ be a power series with non-negative coefficients $a_n$ and radius of convergence 1. Then, $\sum_{n\geq 0}s^n\underset{s\to 1}{\sim} c/(1-s)^\beta$, where $s\in [0,1]$, if and only if $\sum_{k=0}^na_k\underset{n\to \infty}{\sim}c'n^{\beta}$, for postive $\beta$ and where $c$ and $c'$ are constants determining each other. Moreover, if $a_n$ is non-increasing, then one can replace $\sum_{k=0}^na_k$ with $a_n$. See for example Corollary 1.7.3 in Bingham, Goldie and Teugel's book Regular variation.
I'm wondering if the following is true. For two functions $f$ and $g$, write $f\asymp g$ if there exists $C\geq 0$ such that $f\leq Cg$ and $g\leq Cf$. Let $A(z)=\sum a_nz^n$ be a power series with non-negative coefficients $a_n$ and radius of convergence 1. Then, $\sum_{n\geq 0}s^n \asymp 1/(1-s)^\beta$ for $s\in [0,1]$ if and only if $\sum_{k=0}^na_k\asymp n^{\beta}$, if and only if $a_n\asymp n^{\beta}$ (maybe assuming that $a_n$ is non-increasing for the last one).
Note that since we only have information on $\sum s^n$ on the real axis, we cannot use Cauchy integral formula.
EDIT : Also asked on mathoverflow and got a positive answer there.