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.)

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


First hint: $$\sqrt{n+1}-2\sqrt{n}+\sqrt{n-1} = \sqrt{n+1}-\sqrt{n}+\sqrt{n-1}-\sqrt{n}$$
First term:
$$\sqrt{n+1}-\sqrt{n} = (\sqrt{n+1}-\sqrt{n})\frac{\sqrt{n+1}+\sqrt{n}}{\sqrt{n+1}+\sqrt{n}} = \frac{1}{\sqrt{n+1}+\sqrt{n}}$$
Second term:
$$\sqrt{n-1}-\sqrt{n} = (\sqrt{n-1}-\sqrt{n})\frac{\sqrt{n-1}+\sqrt{n}}{\sqrt{n-1}+\sqrt{n}} = \frac{-1}{\sqrt{n-1}+\sqrt{n}}$$
Plug in the first and the second result into the original term:
$$\begin{align}
\frac{1}{\sqrt{n+1}+\sqrt{n}}-\frac{-1}{\sqrt{n-1}+\sqrt{n}}&=\frac{\sqrt{n-1}-\sqrt{n+1}}{(\sqrt{n+1}+\sqrt{n})(\sqrt{n-1}+\sqrt{n})}\\
&=\frac{-2}{(\sqrt{n+1}+\sqrt{n})(\sqrt{n-1}+\sqrt{n})(\sqrt{n+1}+\sqrt{n-1})} \end{align}
$$
Thus $\sqrt{n+1}-2\sqrt{n}+\sqrt{n-1} = O\left(\frac{1}{\sqrt{n}^3}\right)$. This should help you...
