Question

Test the convergence of the series sigma_n = 1^infinity n^n/n!

Answer 1

With $a_n=\frac{n^n}{n!}$, we have

$\dfrac{a_{n+1}}{a_n}=\dfrac{\frac{(n+1)^{n+1}}{(n+1)!}}{\frac{n^n}{n!}}=\dfrac{(n+1)n!}{(n+1)!}\left(\dfrac{n+1}n\right)^n=\left(1+\dfrac1n\right)^n$

whose limit is (famously) $e$ as $n\to\infty$. $e>1$, so the series diverges by the ratio test.