Why does the series converge?

Hello.

How to prove that the series converge?

$\displaystyle \[S = \sum\limits_{n = 2}^{ + \infty } {{{( - 1)}^n} \cdot {c_n}} = \sum\limits_{n = 2}^{ + \infty } {\frac{{{{( - 1)}^n}}}{{\sqrt n + {{( - 1)}^n}}}} \]$

That's obviously an alternating series because $\displaystyle \[{c_n} > 0\]$

In addition $\displaystyle \[\mathop {\lim }\limits_{n \to \infty } {c_n} = 0\]$

But $\displaystyle \[{c_{n + 1}}\]$ is NOT always less than $\displaystyle \[{c_n}\]$

and apparently it prevents me from using Leibniz's test.

Without subtractions $\displaystyle \[\sum\limits_{n = 2}^{ + \infty } {{c_n}} \]$ diverges so I've got no way to go :(

Am I missing something obvious?