Proving convergence of series $\sum_{k=1}^n {\sqrt{k^3+1}-\sqrt {k^3-1}}$ How to prove convergence of the following?
$$\sum_{k=1}^n {\sqrt{k^3+1}-\sqrt {k^3-1}}$$
Thanks!

hint:$$\sum_{k=1}^n (\sqrt{k^3+1}-\sqrt {k^3-1})=\\\sum_{k=1}^n (\sqrt{k^3+1}-\sqrt {k^3-1})\frac{\sqrt{k^3+1}+\sqrt {k^3-1}}{\sqrt{k^3+1}+\sqrt {k^3-1}}
=\\\sum_{k=1}^n \frac{2}{\sqrt{k^3+1}+\sqrt {k^3-1}}\sim \sum_{k=1}^n \frac{2}{2\sqrt{k^3}}$$
