Find a real number $A>0$ such that $\sum_{n=1}^\infty \frac{n!}{(cn)^n}$ converges if $c > A$ and diverges if $c < A$.
I started by performing the ratio test. From that I deduced that $c$ would have to be some sort of fraction for it to satisfy the problem's conditions. However, I'm unsure about how to continue the problem to find the value of $A$. Could I have some help?