Methods of evaluating $ \sum_{k=1}^\infty \frac{(m+k)!}{k!}\frac{1}{5^k}$? I am interested in ways of evaluating the following infinite seris:
$$
\sum_{k=1}^\infty \frac{(m+k)!}{k!}\frac{1}{5^k}.
$$
I already know the answer from Wolfram Alpha but I would like to see some methods of evaluating it as I haven't been able to find many (any?) examples involving an infinite series with the $(m+k)!$ in the numerator and the $k!$ in the denominator, it seems that it is more common to find $k!$ in the numerator and $(m+k)!$ in the denominator.
So what are some methods that can be used to evaluate this series?

This is the non-calculus way. Let the sum be $S_m$. Then,
\begin{align}
& 5S_m-S_m =\sum_{k=0}^{\infty}\frac{(m+k+1)\cdots(k+2)}{5^k}-\sum_{k=1}^{\infty}\frac{(m+k)\cdots(k+1)}{5^k}\\
=&(m+1)!+m\sum_{k=1}^{\infty}\frac{(m+k)\cdots(k+2)}{5^k}\\
=&(m+1)!+5m\sum_{k=2}^{\infty}\frac{(m-1+k)\cdots(k+1)}{5^k}\\
=& (m+1)!+5m\left(S_{m-1}-\frac{m!}{5}\right)=m!+5mS_{m-1}\\
\implies& S_m=\frac{m!}{4}+\frac{5m}{4}S_{m-1}\tag{1}\\
\implies&S_m=\frac{m!}{4}+\frac{5m}{4}\left(\frac{(m-1)!}{4}+\frac{5(m-1)}{4}S_{m-2}\right)\\
=&\frac{m!}{4}+\frac{5m!}{4^2}+\frac{5^2m(m-1)}{4^2}S_{m-2}\\
=&\frac{m!}{4}+\frac{5m!}{4^2}+\frac{5^2m!}{4^3}+\frac{5^3m(m-1)(m-2)}{4^3}S_{m-3}\\
=&\frac{m!}{4}\sum_{k=0}^n\left(\frac{5}{4}\right)^k+\frac{5^{n+1}m\cdots(m-n)}{4^{n+1}}S_{m-n-1}
\end{align}
Setting $n=m-1$ gives us,
$$
S_m=\frac{m!}{4}\sum_{k=0}^{m-1}\left(\frac{5}{4}\right)^k+m!\left(\frac{5}{4}\right)^mS_0
$$
As $S_0$ is a geometric series with value $\frac{1}{4}$ our expression becomes,
$$
S_m=\frac{m!}{4}\sum_{k=0}^{m}\left(\frac{5}{4}\right)^k=m!\left(\left(\frac{5}{4}\right)^{m+1}-1\right)
$$
