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 $\lim\limits_{n \to \infty} a_n=\lim\limits_{n \to \infty} (-1)^n (\sqrt {n^2+n}-n)$
or, $=\lim\limits_{n \to \infty} (-1)^n (\sqrt {n^2+n}-n) \times \dfrac{(\sqrt {n^2+n}-n)}{(\sqrt {n^2+n}+n)}$
or, $=\lim\limits_{n \to \infty} (-1)^n \dfrac{1}{\sqrt {\frac{n^2+n}{n}}+1}$
or, $=\lim\limits_{n \to \infty} (-1)^n \dfrac{1}{2} \ne 0$
Therefore, the given series diverges by the $n$-th Term Test.