# Homework Help: Power Series- radius of convergence

1. Apr 22, 2010

### Roni1985

1. The problem statement, all variables and given/known data
determine the radius of convergence of the given power series

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

2. Relevant equations

3. The attempt at a solution
I did the ratio test
then I had to take the 'ln'
|e|<1 for the series to converge.
It never happens but according to the answers the radius is 'e'.

2. Apr 22, 2010

### Roni1985

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

3. Apr 22, 2010

### 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.