The rate of convergence of the remainder of the power series for the Polylog function

Question

Let $0<p<1$ be a positive real number strictly smaller than one and $q>0$ be a positive real number. Consider the series $$ \mathsf{Li}_{-q}(p) = \sum_{\ell=1}^{+\infty}\ell^{q}p^{\ell} $$ which defines the polylogarithmic function. Is there any result on the rate at which the remainder $$ c_n=\sum_{\ell=n}^{+\infty}\ell^{q}p^{\ell} $$ goes to zero? My guess is $c_n=O(n^q\,p^n)$, but I cannot find anything on this topic.

Answer 1

$$\frac{c_n}{n^q p^n}\underset{\ell=n+k}{\quad=\quad}\sum_{k=0}^\infty p^k\left(1+\frac kn\right)^q\quad\color{blue}{\underset{n\to\infty}{\longrightarrow}}\quad\sum_{k=0}^\infty p^k=\frac1{1-p}$$ by Tannery's theorem (with $\sum_{k=0}^\infty p^k(1+k)^q$ being the dominating convergent series).