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