# Thread: Test for Series Convergence/Divergence

1. ## Test for Series Convergence/Divergence

Hi all, I have a series sum from n = 1 to infinity for n!/n^n.

I've used the ratio test only to come up with a 1 which is obviously inconclusive. I tried the root test and also came up with a 1 though it's quite possible I mixed this up. I feel as though it has to do with either direct comparison or limit comparison but I'm drawing blanks and any help would be greatly appreciated.

Zo

2. Originally Posted by zo1971so
Hi all, I have a series sum from n = 1 to infinity for n!/n^n.

I've used the ratio test only to come up with a 1 which is obviously inconclusive. I tried the root test and also came up with a 1 though it's quite possible I mixed this up. I feel as though it has to do with either direct comparison or limit comparison but I'm drawing blanks and any help would be greatly appreciated.

Zo
let me guess ...

you did the ratio test and came up with

$\displaystyle \lim_{n \to \infty} \left(\frac{n}{n+1}\right)^n$

and you think the limit is 1 , right?

3. I didn't even recognize it as 1/e with it written as (n^n)/(n+1)^n

Thanks a ton.

Edit 2: You beat me to it, surprise = 0

4. Originally Posted by zo1971so
I didn't even recognize it as e with it written as (n^n)/(n+1)^n

Thanks a ton.
actually, the limit is $\displaystyle \frac{1}{e}$ ...

5. For those of us who are a little slow (i.e., me), I've written out a detailed proof

We can use the ratio test to test for this series' convergence.

$\displaystyle L=\lim_{n\to\infty}\frac{\frac{\left(n+1\right)!}{ \left(n+1\right)^{n+1}}}{\frac{n!}{n^n}}$
$\displaystyle =\lim_{n\to\infty}\frac{\left(n+1\right)!}{\left(n +1\right)^{n+1}}\frac{n^n}{n!}$
$\displaystyle =\lim_{n\to\infty}\frac{\frac{n+1}{n^n}}{\left(n+1 \right)^{n+1}}$
$\displaystyle =\lim_{n\to\infty}\frac{n^n}{\left(n+1\right)^n}$
$\displaystyle =\lim_{n\to\infty}\left(\frac{n}{n+1}\right)^n$
$\displaystyle =\lim_{n\to\infty}\left(\frac{1}{\left(n+1\right)/n}\right)^n$
$\displaystyle =\frac{1}{e}$

The ratio test specifies that if $\displaystyle L<1$ the series converges absolutely. Thus, this series converges absolutely.

http://maths-help.info/questions/10