How to check the convergence of $\sum_{n=1}^{\infty} \frac{n^{n/2}}{n!}$

Question

How do I know the nature of the

$$\begin{align} \sum_{n=1}^{\infty} \frac{n^{n/2}}{n!} \end{align}$$

I have tried the ratio test and got nowhere.

Answer 1

\begin{align} \frac{(n+1)^{\frac{n+1}2}}{(n+1)!}\cdot \frac{n!}{n^\frac{n}2} &= \frac{\sqrt{n+1}}{n+1}\sqrt{\left( \frac{n+1}{n}\right)^n}=\frac{\sqrt{n+1}}{n+1}\sqrt{\left( 1+\frac{1}{n}\right)^n} \end{align}

I am going to leave the evaluation of the limit as $n\to \infty$ as an exercise.