Answer
Diverges
Work Step by Step
The $n$-th Term Test states that when a series $a_n \to 0$ then the series diverges.
We notice that $\lim\limits_{n \to \infty} a_n=\lim\limits_{n \to \infty} (-1)^n \dfrac{n}{n+1}$
a) When $n$ is even, then $\lim\limits_{n \to \infty} a_n=\lim\limits_{n \to \infty} \dfrac{n}{n+1}=\lim\limits_{n \to \infty} \dfrac{n}{1+1/n}=\dfrac{1}{1+0}=1$
b) When $n$ is odd, then $\lim\limits_{n \to \infty} a_n=\lim\limits_{n \to \infty} -\dfrac{n}{n+1}=\lim\limits_{n \to \infty} -\dfrac{n}{1+1/n}=-\dfrac{1}{1+0}=-1$
Therefore, the given series Diverges by the $n$-th Term Test.