# Limit of series

1. Aug 20, 2008

### ercagpince

1. The problem statement, all variables and given/known data
what is the limit of this expression?

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

2. Relevant equations

3. 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.

2. Aug 20, 2008

### Dick

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

3. Aug 20, 2008

### ercagpince

Could you show it explicitly?

4. Aug 20, 2008

### Dick

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?

5. Aug 20, 2008

### ercagpince

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

6. Aug 20, 2008

### Dick

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

7. Aug 20, 2008

### ercagpince

thanks for the post