# Absolute convergence: ratio/root test n!/n^n

1. Nov 24, 2009

### SpicyPepper

1. The problem statement, all variables and given/known data
Doing some problems from textbook, I need to determine whether the series is absolutely convergent, conditionally convergent, or divergent.

n!/n^n

I plugged it into WA, and it says the series doesn't converge, but I'm not sure how to figure it out.

2. Relevant equations

3. The attempt at a solution

First, I applied the root test

lim n->inf $$\frac{(n+1)!}{(n+1)^n} * \frac{n^n}{n!}$$

lim n->inf $$\frac{(n+1)n!}{(n+1)(n+1)^n} * \frac{n^n}{n!}$$

I reduce this, and apply the root test:

lim n->inf $$\sqrt[n]{\frac{n^n}{(n+1)^n}}$$

lim n->inf $$\frac{n}{n+1}$$

lim n->inf $$\frac{1}{1 + 1/n}$$

= 1

1 means that it's inconclusive. I'm not sure if I applied the tests incorrectly or if I'm supposed to try something else.

2. Nov 24, 2009

### Billy Bob

It seems you took the root of the ratio. That's wrong. Don't combine the two tests. Use one or the other.

The ratio test will work. (Your first step has a typo, but the second step has fixed it.) To finish it off, observe

$$\frac{n^n}{(n+1)^n}=\frac{1}{\left( \frac{n+1}{n} \right)^n}=\frac{1}{\left( 1+\frac{1}{n} \right)^n}$$.

The last expression has a famous limit.

Actually the series converges, and as a double check using the comparison test, it is less than 2/n^2.

3. Nov 24, 2009

### SpicyPepper

Thanks for mentioning the typos, I see them. I meant to say I applied the ratio test first, and the exponent in the denominator of my first line should be n+1.

I remember the limit from deriving it with L'Hopital's rule, 1/e. Thanks, I simply didn't see I could reduce it by dividing by n^n/n^n.