Find the sum of the series $\sum_{n=1}^{\infty}(\sqrt{n+2}-2\sqrt{n+1}+\sqrt{n})$ I need to test the convergence and find the sum of the following series:
$\sum_{n=1}^{\infty}(\sqrt{n+2}-2\sqrt{n+1}+\sqrt{n})$
But i am not really sure what kind of series is this?
Since 
$2\sqrt{n+1}>\sqrt{n}+\sqrt{n+2} \forall n \in \mathbb{N}$
It means that this is the series with all negative terms, what can i do with that kind of series? Can i use the same tests as for series with all positive terms (non-negative to be precise)? If that is true, then how can i use them?

Since
\begin{align*}
\sqrt{n+2}-2\sqrt{n+1}+\sqrt{n}&=\sqrt{n+2}-\sqrt{n+1}-(\sqrt{n+1}-\sqrt{n})\\
&=\frac{1}{\sqrt{n+2}+\sqrt{n+1}}-\frac{1}{\sqrt{n+1}+\sqrt{n}}
\end{align*}
Then, the sum is telescopic:
\begin{align*}
\sum_{n=1}^N(\sqrt{n+2}-2\sqrt{n+1}+\sqrt{n})&=\frac{1}{\sqrt{N+2}+\sqrt{N+1}}-\frac{1}{\sqrt{2}+1}\\
\sum_{n=1}^{\infty}(\sqrt{n+2}-2\sqrt{n+1}+\sqrt{n})&=-\frac{1}{\sqrt{2}+1}\\
&=-(\sqrt{2}-1)
\end{align*}
