Trying to show that $\sum_{1}^{\infty} \frac{n^n}{n!} $ diverges. I have been trying to prove this using the ratio test $|\frac{A_{n+1}}{A_n}|$ , which leads me to this expression: $$\left|\frac{(n+1)^{n+1}}{(n+1)!}\cdot \frac{n!}{n^n}\right|=\left|\frac{(n+1)^{n+1}}{n^n\cdot(n+1)}\right|\cong as \space n\to\infty \frac{n^{n+1}}{n^{n+1}}$$ While it is true that the series diverges, I am getting the wrong ratio value.

Hint: Show by induction or inspection that $n^n \geq n!$. Then your terms do not converge to zero so the series necessarily diverges.
