# Does the denominator become larger faster

1. Mar 19, 2013

### whatlifeforme

1. The problem statement, all variables and given/known data
Determine if convergent or divergent. Determine limit if convergent.

2. Relevant equations
$a_{n} = \frac{n!}{n^n}$

3. The attempt at a solution
As per the hint, i use 1/n to compare.

however, how is this statement true:

$\lim_{x\to\infty} \frac{n!}{n^n} <= \lim_{x\to\infty} \frac{1}{n}$??

does the denominator become larger faster than the numerator making it a smaller number than the 1/n?

2. Mar 19, 2013

### Dick

Think about it. E.g. 3!/3^3 is less than 1/3. Why is that? That's (1*2*3)/(3*3*3).

3. Mar 21, 2013

### whatlifeforme

update: nevermind.