Use the Ratio Test.

$\lim\limits_{n\to\infty}\Bigg|\frac{\frac{x^{n+1}}{n+1}}{\frac{x^n}{n}}\Bigg|=$

$\lim\limits_{n\to\infty}|x|\cdot\frac{n}{n+1}=$

$|x|<1$

If $x = -1$:

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

Does this resultant series converge or diverge?
Be sure to check the resultant series if $x = 1$.