sequences and series - Convergence using Root Test

Problem: test if the series converges

$$\sum_{n=1}^ \infty \frac {(-2)^{n+1}} {n^{n+1}}$$

My approach:

I see it is equal to

$$\sum_{n=1}^ \infty \frac {(-2)^n} {n^n} \cdot \frac {-2} n$$ , and $\sum_{n=1}^ \infty \frac {(-2)^n} {n^n}$ converges absolutely using root test, and $\sum_{n=1}^ \infty \frac {-2} n$ diverges by using p-series test.

So is the original series divergent because convergent * divergent = divergent?

Is convergent * convergent = convergent??