Answer
The series diverges by the Test for Divergence.
Work Step by Step
Use the test for divergence.
$\lim\limits_{n\to \infty}|a_n|=\lim\limits_{n\to \infty}\left|(-1)^n\frac{n}{n+2}\right|=\lim\limits_{n\to \infty}\frac{n}{n+2}=1$. Thus $\lim\limits_{n\to \infty}(-1)^n\frac{n}{n+2}$ does not exit, so the series is divergent by the Test for Divergence.