Limit of $\frac{N^{n}(N!)^{n}}{(n-1)!}$ as $n$ tends to infinity

Question

How to prove that $$\lim_{n\rightarrow\infty}\frac{N^{n}(N!)^{n}}{(n-1)!}=0?$$

Any help would be greatly appreciated.

Answer 1

By ratio test we have

$$ \frac{N^{n+1}(N!)^{n+1}}{(n)!}\frac{(n-1)!}{N^{n}(N!)^{n}}=\frac{N\, N!}{n}\to 0$$

Answer 2

Your numerator is $\alpha^n$, where $\alpha=N.N!$. So, we have the limit of $\frac{\alpha^n}{(n-1)!}$, which is $0$; since $\alpha$ is constant.

Answer 3

$$\lim_{n\rightarrow\infty}\frac{N^{n}(N!)^{n}}{(n-1)!}=\lim_{n\rightarrow\infty}\frac{M^{n+1}}{n!}=\frac{M^{M}}{M!}\lim_{n\rightarrow\infty}\frac{M\cdot M\cdot M\cdots M}{(M+1)\cdot(M+2)\cdots(M+n)} \\<\frac{M^{M}}{M!}\lim_{n\rightarrow\infty}\left(\frac M{M+1}\right)^n.$$