Convergence of the function series $\sum _{n=1}^{\infty}\dfrac{(nx)^n}{n!}$

Question

Determine $x \in \mathbb{R}$ such that the series $$\sum _{n=1}^{\infty} \dfrac{(nx)^n}{n!}$$ converges punctually.

I tried everything, but I'm really stucked. The book says that the series converges for $0 \le x <e^{-1}$. I tried with the ratio test, the infinitesimal test, I tried to do also $\lim _{n\rightarrow +\infty} \dfrac{(nx)^n}{n!}$ to see where the succession is infinitesimal, but I cannot figure it out.