Question

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

Answer 1

With
`a_n=(n^n)/(n!)`, we have

`(a_(n+1))/(a_n)=(((n+1)^(n+1))/((n+1)!))/((n^n)/(n!))=((n+1)n!)/((n+1)!)\left(\frac{n+1}n\right)^n=\left(1+\frac1n\right)^n`

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