summation:

$\displaystyle \sum\limits_{n=0} ^\infty \frac{(-1)^{n+1}n^n}{3^n^2}$

Using the root test and dropping the $\displaystyle (-1)^{n+1}$ I got:

$\displaystyle ^n\sqrt{\frac{n^n}{3^n^2}}$

simplifies to:

$\displaystyle \frac{n}{3^n}$

To me it appears that any positive integer used will give the result of a ratio value of less than 1 and thus converges. However, wolfram tells me that it actually diverges. What am I doing wrong??