real analysis - For what values of $x$ does the series converge: $sum limits_{n=1}^{infty} frac{x^n}{n^n}$?

For what values of $x$ do the following series converge or diverge

$$\sum \limits_{n=1}^{\infty} \frac{x^n}{n^n}$$

I tried to solve this using the ratio test where the series converge when

$$\lim \limits_{n \to \infty} \frac{x^{n+1}n^n}{(n+1)^{n+1}x^n} <1$$

$$\lim \limits_{n \to \infty} \frac{xn^n}{(n+1)^{n+1}} <1$$

but then I am not sure what to do next.

Please give me some ideas or hints on how to solve this question, thanks to anybody who helps.

Answer

Staring from the last step you have shown we have
$$\lim \limits_{n \to \infty} \frac{xn^n}{(n+1)^{n+1}} <1.$$
But
$$\lim \limits_{n \to \infty} \frac{xn^n}{(n+1)^{n+1}} =\lim_{n \to \infty} \frac{x}{\left(1+\frac{1}{n}\right)^n(n+1)}.$$
As $n \to \infty$, the limit $\lim_{n \to \infty}\left(1+\frac{1}{n}\right)^n=e.$ Thus the limit
$$\lim_{n \to \infty} \frac{x}{\left(1+\frac{1}{n}\right)^n(n+1)}=0 \quad \forall x.$$