Power Series- radius of convergence

[Roni1985]

Homework Statement

determine the radius of convergence of the given power series

$$\sum$$$$^{inf}_{n=1}$$$$\frac{n!x^n}{n^n}$$

Homework Equations

The Attempt at a Solution

I did the ratio test
then I had to take the 'ln'
but, my answer is this
|e|<1 for the series to converge.
It never happens but according to the answers the radius is 'e'.

 

[Roni1985]

oh I think I got it, I didn't have to lhopital lnx in the middle ...
:\

 

[Hoblitz]

Applying the Ratio Test, we have the sequence

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

Taking the limit as n goes to infinity... this might look like a familiar limit. Then recall that the Radius of convergence is the reciprocal of this limit.