Asymptotics of Geometric Distribution Moments

Question

It is known (e.g., see Wikipedia) that the $k$th moment of a Geometric distribution with success probability $p$ is $$\mathbb{E}\left[X^k\right] = \sum_{j = 0}^\infty j^k \cdot p(1-p)^j = p \cdot\text{Li}_{-k}(1-p)$$ I'm wondering if for a fixed $p$, can we find nicer asymptotic bounds as $k \to \infty$ for this value, say, in terms of just elementary functions? If it helps, we may assume that $p$ is sufficiently far from both $0$ and $1$, e.g. $p = 1/e$.