How can one determine whether the following series converges or diverges $$
\sum_{n = 2}^{\infty}\frac{(-1)^{n}}{\sqrt{n} + (-1)^{n}}
$$

Wolfram Alpha returns nothing useful, except that the ratio test was inconclusive.
If 
$$
a_n=\frac{(-1)^n}{(-1)^n+\sqrt{n}},
$$
then
$$
a_{2n}+a_{2n+1}=\frac{1}{\sqrt{2n}+1}-\frac{1}{\sqrt{2n+1}-1}=
\frac{\sqrt{2n+1}-\sqrt{2n}-2}{(\sqrt{2n+1}-1)(\sqrt{2n}+1)}=-\frac{1}{n}+{\mathcal O}(n^{-3/2}).
$$
Hence, the series diverges to $-\infty$.
One way of doing it, which is not necessarily the most sophisticated way to go about it, is to look at the partial sums and see if those trend toward some limit. Let's, for the sake of clear example, use the sum "$1 + 1/2 + 1/4 + 1/8 + ...$" if we add the first two terms, our partial sum is 1.5. For the first three it's 1.75. And for the first 4 it's 1.875. As you can seem the partial sums are trending toward two and if we had a function which gave us these partial sums it's limit as x approaches infinity would be two as well.