I want to show that the series
$\sum_{n\geq 1}\frac{1}{(n+1)^{a+1}}\sum_{k=0}^n b^k\left(\frac{(n-k)!}{n!}\right)^a$
converges for $a,b>0$. I have tried this so much that the smallest hint will probably suffice. I asked a question before which would have been enough but it is not true. Right now I am really stuck and frustrated. Any help would be greatly appreciated!