$\sum_{n=0}^\infty(-1)^n(\sqrt{n^5}-\sqrt{n^5-n^2})$
My attempt: I tried to use Leibniz's test and found that a_n approaches 0 when sent to infinity, but I cannot tell if it monotonic decreases as my attempt to differentiate it ended in horrendus failure.
$\sqrt {n^{5}} -\sqrt {n^{5}-n^{2}}=\frac {n^{5} -({n^{5}-n^{2})}} {{n^{5/2}} +\sqrt {n^{5}-n^{2}}}=\frac {n^{2}} {{n^{5/2}} +\sqrt {n^{5}-n^{2}}}=\frac 1 {n^{1/2}+\sqrt {n-n^{-2}}}$. Now $x-\frac 1 {x^{2}}$ is increasing and non-negative on $[1,\infty)$. So its square root is also increasing. It is now clear that ${\sqrt {n^{5}} -\sqrt {n^{5}-n^{2}}}$ is decreasing and tends to $0$. By alternating series test the series is conergent. It is not absolutely convergent by comparison with $\sum n^{-1/2}$.