Find the radius of convergence of the following series

$\displaystyle \sum_1^\infty \frac{n!}{n^n} t^n$

Using the ratio test I get the following:

$\displaystyle |t| \lim (n\to\infty) |\frac{n+1}{n \cdot n!}$

As far as I can see this converges for all real numbers, but the answer says

$\displaystyle -e<t<e$

Where am I going wrong? Thanks.