$$\sum\_{n=1}^{\infty} (-1)^n(\sqrt{n+1}-\sqrt{n})$$

I know that $\sum\_{n=1}^{\infty} \sqrt{n+1}-\sqrt{n}$ diverges since the kth partial sum of the telescoping series can be written as:

$(\sqrt{2}-\sqrt{1})+(\sqrt{3}-\sqrt{2})+(\sqrt{4}-\sqrt{3})+...+(\sqrt{k+1}-\sqrt{k})=\sqrt{k+1}-1$

Then taking the limit as $k\rightarrow\infty$ of the kth partial sum gives me

$\lim\_{n\rightarrow\infty}=\sqrt{k+1}-1=\infty$

So this series is not absolutely convergent. But how can I check if it is conditionally convergent?

Intuitively I'm thinking if $\sqrt{n+1}-\sqrt{n}$ diverges then
$\sum\_{n=1}^{\infty} (-1)^n(\sqrt{n+1}-\sqrt{n})$ must diverge too. Is my logic correct here?

The solution says that this series is conditionally convergent but I'm not sure how they get that? Do I need to use the Alternating series test?