<s>[INST] 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!
 [/INST] $\sqrt{k^3+1}-\sqrt {k^3-1}=\frac{(\sqrt{k^3+1}-\sqrt {k^3-1})(\sqrt{k^3+1}+\sqrt {k^3-1})}{(\sqrt{k^3+1}+\sqrt {k^3-1})}=\frac{2}{\sqrt{k^3+1}+\sqrt {k^3-1}}\leq \frac{2}{\sqrt{k^3+1}}\leq \frac{2}{\sqrt{k^3}}=\frac{2}{k^{1+\frac{1}{2}}}$
Then apply comparison test.
</s>