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.

Note that
$$
n^n \ge n! \implies \frac{x^n}{n^n} \le \frac{x^n}{n!}
$$
and use the comparison test against the series formulation of $e^x$

Note the above only works for positive $x$, however consider showing that the series is absolutely convergent for any $x$ using the above inequalities and thus it is convergent for any $x$.
