I first posted this on mathematics. However, I got no answer there and it seems adapted here too. Also, it seems to be harder than I first thought.

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. Let $\beta>0$. Then, as the **real** variable $s\in [0,1]$ tends to 1, $\sum_{n\geq 0}a_ns^n\underset{s\to 1}{\sim} c/(1-s)^\beta$ if and only if $\sum_{k=0}^na_k\underset{n\to \infty}{\sim}c'n^{\beta}$, where $c$ and $c'$ are determine each other. Moreover, if $a_n$ is non-increasing, then one can replace $\sum_{k=0}^na_k$ with $a_n$, that is, we have $a_n\sim c''n^{\beta-1}$. 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, $\sum_{n\geq 0}a_ns^n \asymp 1/(1-s)^\beta$ for $s\in [0,1]$ if and only if $\sum_{k=0}^na_k\asymp n^{\beta}$ if and only if $a_n\asymp n^{\beta-1}$. The implicit constants are asked not to depend on $s$ and $n$ respectively.