Limit of $\frac{a^{n+1}(n+1)!^b}{\sum_{k=0}^n a^kk!^b}$ when $n\rightarrow\infty$

Question

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!

Answer 1

Without further conditions on $a$ and $b$, this cannot be true. For example, let $a=b=1$, and let $n \ge 1$. Then the numerator is $(n+1)!$. The denominator is $\le n(n-1)!+n!$, that is, $\le 2(n!)$, so the ratio is $\ge (n+1)/2$.