With [tex]a_n=\frac{n^n}{n!}[/tex], we have
[tex]\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[/tex]
whose limit is (famously) [tex]e[/tex] as [tex]n\to\infty[/tex]. [tex]e>1[/tex], so the series diverges by the ratio test.
