\[lim_{n \rightarrow \infty} \sum^n_{k=1}\frac{k!}{k^k} = \frac{1!}{1^1} + \frac{2!}{2^2}+...+ \frac{n!}{n^n}\]
diverge or converge?By the ratio test,
\[\frac{a_{n+1}}{a_n} = \frac{(n+1)!/(n+1)^{n+1}}{n!/n^n}= \frac{n^n}{(n+1)^n}\]
.\[\begin{equation} \begin{aligned} lim_{n \rightarrow \infty} \frac{n^n}{(n+1)^n} &= lim_{n \rightarrow \infty} \frac{n^n}{(n(1+1/n))^n} \\ &= lim_{n \rightarrow \infty} \frac{n^n}{n^n(1+1/n)^n} \\ &= lim_{n \rightarrow \infty} \frac{1}{(1+1/n)^n} \\ &= \frac{1}{e} \lt 1 \end{aligned} \end{equation}\]
The sequence converges by the ratio test.