$$\sum\_{n=1}^{\infty}\frac{2^nn!}{n^n}$$

My attempt using root test:

$$\lim\_{n \to{\infty}}\sqrt[n]{\frac{2^nn!}{n^n}}=\lim\_{n \to{\infty}}2\sqrt[n]{\frac{n!}{n^n}}$$
It can easily be proved that for positive $n$ we have $n^n>n!$. Now what should I do with this limit?

Note: I want to solve the problem with root test not other tests.