Answer
Diverges
Work Step by Step
Consider the $n$-th Term Test for a series $u_n$; when $u_n \to 0$ then the series diverges.
From the series, we notice that $\lim\limits_{n \to \infty} u_n=\lim\limits_{n \to \infty} (-1)^n (\sqrt {n^2+n}-n) \\=\lim\limits_{n \to \infty} (-1)^n (\sqrt {n^2+n}-n) \times \dfrac{(\sqrt {n^2+n}-n)}{(\sqrt {n^2+n}+n)} \\=\lim\limits_{n \to \infty} (-1)^n \dfrac{1}{\sqrt {\dfrac{(n^2+n)}{n}}+1} \\ =\lim\limits_{n \to \infty} (-1)^n (\dfrac{1}{2}) \ne 0$
This implies that the given series Diverges by the $n$-th Term Test.