does this series converge? $\sum_{n=1}^\infty{\left( \sqrt[3]{n+1} - \sqrt[3]{n-1} \right)^\alpha} $ 
show the the following series converge\diverge
$\sum_{n=1}^\infty{\left( \sqrt[3]{n+1} - \sqrt[3]{n-1} \right)^\alpha} $

all the test i tried failed (root test, ratio test,direct comparison)
please dont use integrals as this is out of the scope for me right now

Hint rationalize:
$$\left( \sqrt[3]{n+1} - \sqrt[3]{n-1} \right)^\alpha = \left( \frac{2}{(\sqrt[3]{n+1})^2+ \sqrt[3]{n-1}\sqrt[3]{n+1}+ (\sqrt[3]{n-1})^2} \right)^\alpha$$
Compare with $$\sum\frac{1}{n^{\frac{2\alpha}{3}}}$$
