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