Infinite series : convergence.

$\displaystyle \sum_{n=1}^{+\infty} \frac{2^nn!}{n^n}$. I must show whether it converges or not. Using my intuition I think it converges. I tried to use the root test, the only test I think that works.

So I'm stuck when I must find the limit when $\displaystyle n$ tends to $\displaystyle +\infty$ of $\displaystyle \frac{2\cdot n!^{\frac{1}{n}}}{n}$. Does the limit of $\displaystyle n!^\frac{1}{n}=\frac{1}{e}$, when $\displaystyle n$ tends to $\displaystyle +\infty$? If yes, how to prove it using maths of calculus II?