If I have the series:

$$\sum\_{n=1}^{\infty} (-1)^n \frac{n}{n+2}$$

I need to know if this series converges or diverges. The solution says to use the n-th term test for divergence. In general I know that means that if you take $\lim\_{n\rightarrow\infty} a\_{n}$ and get any number other than 0, it diverges. However, I'm confused here because of the $(-1)^n$

The solution says to do

$\lim\_{n\rightarrow\infty}\frac{n}{n+2}$ and this diverges by n-th term test since it equals $1$. However, can I just do this anytime I have something that has alternating $(-1)^n$? Just apply the n-th term test to the part that doesn't have the $(-1)^n$ and if that diverges then the whole thing diverges?