Behavior of $\sum_{k=p}^\infty k^p a^k$ Let $a\in (-1,1)$, $p\in\mathbb{N}_{>0}$, and consider the series
$$
F(a,p) = \sum_{k=p}^\infty k^p a^k
$$

$F(a,p)$ is always convergent since $|a|<1$, but is it possible to characterize its behavior as a function of $a$, $p$?

Symbolic math routines return the expression:
$$
F(a,p) = a^p \Phi(a,-p,p), 
$$
where $\Phi$ is the Hurwitz-Lerch transcendent function. After some web searching, I could not find a simple description of the asymptotic properties of this function. My question may be trivial and I apologize if this is the case. Any comment or reference is welcome. Thanks.
 A: We have $k^p \geq p!\binom k p$, therefore for positive $a$, we have $$\begin{align}F(a, p) &\geq p!\sum_{k = p}^\infty \binom k p a^k \\&= p! a^p \sum_{k = 0}^\infty \binom {k + p} p a^k \\&=\frac{p!a^p}{(1-a)^{p+1}}\end{align}$$ which grows more than exponentially in $p$.
A: Partial answer. We can write
$$
F(a,p) = \sum\limits_{k = p}^\infty  {k^p a^k }  = p^p a^p \sum\limits_{n = 0}^\infty  {\left( {1 + \frac{n}{p}} \right)^p a^n } 
$$
Now
$$
\left| {\left( {1 + \frac{n}{p}} \right)^p a^n } \right| \le e^n \left| a \right|^n 
$$
for $p>0$ and $n\geq 0$. Thus, if $\left| a \right| < \frac{1}{e}$, Tannery's theorem gives
$$
\mathop {\lim }\limits_{p \to  + \infty } \frac{{F(a,p)}}{{p^p a^p }} = \sum\limits_{n = 0}^\infty  {\mathop {\lim }\limits_{p \to  + \infty } \left( {1 + \frac{n}{p}} \right)^p a^n }  = \sum\limits_{n = 0}^\infty  {e^n a^n }  = \frac{1}{{1 - ea}}.
$$
Therefore
$$
F(a,p) \sim \frac{{p^p a^p }}{{1 - ea}}
$$
with any fixed $\left| a \right| < \frac{1}{e}$ as $p\to +\infty$.
