Convergence of $\displaystyle\sum_{n=1}^{\infty}\frac{(2i)^{n}\cdot n!}{n^{n}}$

I have to show the series$$\sum_{n=1}^{\infty}\frac{(2i)^{n}\cdot n!}{n^{n}}$$converges. I know it does and I tried to use the ratio test, but in the final limit, I got$$\lim_{n\to\infty}2i\left[\left(1+\frac{1}{n}\right)\right]^{-1}$$which results at$$\frac{2i}{e}$$and I don't know if I can't say it's smaller than 1 because of the imaginary unity.