Does the denominator become larger faster

[whatlifeforme]

Homework Statement

Determine if convergent or divergent. Determine limit if convergent.

Homework Equations

$a_{n} = \frac{n!}{n^n}$

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?

 

[Dick]

Homework Statement

Determine if convergent or divergent. Determine limit if convergent.

Homework Equations

$a_{n} = \frac{n!}{n^n}$

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?

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

 

[whatlifeforme]

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

update: nevermind.