testing for convergence/divergence

I've got a few problems here I'm wanting some guidance over.

1. $\displaystyle \sum_{n=0}^{\infty} \frac{n!}{2^{n^2}}$

Am I right in thinking the Ratio test is the one to use here?

I get

$\displaystyle \frac {n+1}{2^{2n+1}} $

Do I use the Ratio Test again? I end up with $\displaystyle \frac {1}{2}$

2. $\displaystyle \sum_{n=1}^{\infty} \frac{2^n +1}{3^n -cosn}$

Absolute convergence test?

|cos n|<1, thus the above series can be reduced down to $\displaystyle (\frac {2}{3})^n = \frac{2}{3}$

>>edit: sorry, made a mistake in the original post in the series equation. Now rectified.<<

3. $\displaystyle \sum_{n=1}^{\infty} (-1)^{n-1(n^\frac{1}{n}-1)^n}$

I'm thinking the Root test here?

then, since $\displaystyle \lim_{n \to\infty} {n}^\frac{1}{n} = 1$, we're left with just $\displaystyle (-1)^n$

Any/all of these on the right track or have I totally lost the plot here?

Cheers!