# Series either converges or diverges

1. Mar 31, 2009

### v0id19

1. The problem statement, all variables and given/known data
Determine whether the series $$\sum_{n=1}^{\infty}{\frac{n!}{n^n}}$$ converges or diverges.

2. Relevant equations
The Comparison Test
The Limit Comparison Test

3. The attempt at a solution
I know it diverges, and i tried $$a_n=\frac{n!}{n^n}$$ and $$b_n=\frac{1}{n^n}$$ for the limit comparison test, but it gave me infinity which is useless. I also tried the comparison test saying $$a_n=\frac{n!}{n^n} \ge b_n=\frac{1}{n^n}$$ but i don't know how to prove that $$\frac{1}{n^n}$$ diverges

2. Mar 31, 2009

### rwisz

Re: Converge/Diverge

Have you learned of the Ratio Test yet? It would simplify this problem greatly.

Otherwise, just to show that $$\frac{1}{n^{n}}$$ diverges, that can be re-written as $$n^{-n}$$ which would not equate to zero using the nth term test for divergence, therefore it DOES diverge.

I haven't looked at your other work yet, but if all you needed was to prove that 1/n^n diverges then there ya go!

Last edited: Apr 1, 2009