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

Ratio test is inconclusive, but using Raabe's test we can see that the series converges when $\alpha>\frac{3}{2}$.
