Proving the Limit of an Expression: \frac{n^n}{n!}

[ercagpince]

Homework Statement

what is the limit of this expression?

$$lim_{n\rightarrow\infty}\frac{n^n}{n!}$$

Homework Equations

The Attempt at a Solution

I tried to make it look like $$\frac{x^n}{n!}$$ and also tried to apply the sandwich theorem, but got nothing logical.
Probably the limit is $$\infty$$, still I want to prove it mathematically.

 

[Dick]

You could take the log and use Stirling's approximation on the factorial.

 

[ercagpince]

Could you show it explicitly?

 

[Dick]

Could you show it explicitly?

If you mean show the details, Isn't that your job? Did you look up Stirling's approximation? Or do you mean do it without Stirling's formula?

 

[ercagpince]

if you take the stirling's approximation there is no need to take log of the term on numerator.
that why i asked that.

 

[Dick]

if you take the stirling's approximation there is no need to take log of the term on numerator.
that why i asked that.

Right, if you use the form n!~(n/e)^n. I was thinking of ln(n!)~n*ln(n)-n

 

[ercagpince]

thanks for the post