For which values of $\alpha \in \mathbb{R}$, does the series $\sum_{n=1}^\infty n^\alpha(\sqrt{n+1} - 2 \sqrt{n} + \sqrt{n-1})$ converge? How do I study for which values of $\alpha \in \mathbb{R}$ the following series converges?
(I have some troubles because of the form [$\infty - \infty$] that arises when taking the limit.)

$$\sum_{n=1}^\infty n^\alpha(\sqrt{n+1} - 2 \sqrt{n} + \sqrt{n-1})$$


Hint: $\sqrt{n+1}-2\sqrt{n}+\sqrt{n-1} = \frac{-2}{(\sqrt{n}+\sqrt{n+1})(\sqrt{n-1}+\sqrt{n+1})(\sqrt{n}+\sqrt{n-1})}$
So, this term is for big n approximately $-2n^{-\frac{3}{2}}$
