There is some confusion, your title reads series but your post reads sequences... i'm going to assume you want series behavior.
a)Convergent. It is an alternating series, $\displaystyle a_n \to 0 $ as $\displaystyle n \to \infty $
b) Divergent $\displaystyle a_n \to -1 $ as $\displaystyle n \to \infty $ , its like adding -1 infinite times
c) divergent but a bit harder to show, multiply numerator and denominator by the conjugate. consider the following...
$\displaystyle a_n \ = \ \frac{(\sqrt{n + 1} - \sqrt{n}) (\sqrt{n + 1} + \sqrt{n}) }{ \sqrt{n + 1} + \sqrt{n}} $
$\displaystyle a_n \ = \ \frac{1}{\sqrt{n + 1} + \sqrt{n}} $
Now compare this with $\displaystyle b_n \ = \ \frac{1}{3 \sqrt{n}} $ which diverges because the exponent on n is less than 1
$\displaystyle a_n > b_n $ for n > 0 so a_n diverges because every nth term of a_n is LARGER than every corresponding nth term of b_n