I want to prove that $\lim_{n\rightarrow\infty}\frac{a^{n+1}(n+1)!^b}{\sum_{k=0}^n a^kk!^b}<\infty$ for $a,b>0$.
This is the last step of a bigger problem. I believe it would suffice to use good enough upper and lower bounds for the factorials, but I don't know such bounds. Any help would be greatly appreciated!