Apply the Ratio Test to ∑n=1∞n!nn\sum_{n=1}^{\infty} \frac{n!}{n^n}∑n=1∞nnn!.
What is limn→∞∣an+1an∣\lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right|limn→∞anan+1 and what can be concluded?
The limit is 1e\frac{1}{e}e1, so the series converges.
The limit is eee, so the series diverges.
The limit is 111, so the Ratio Test is inconclusive.
The limit is ∞\infty∞, so the series diverges.