$\sum_1^n 2\sqrt{n} - \sqrt{n-1} - \sqrt{n+1} $ converge or not? how to check if this converge? $$\sum_{n=1}^\infty a_n$$
$$a_n = 2\sqrt{n} - \sqrt{n-1} - \sqrt{n+1}$$
what i did is to show that: 
$$a_n =2\sqrt{n} - \sqrt{n-1} - \sqrt{n+1} > 2\sqrt{n} - \sqrt{n+1} - \sqrt{n+1} = 2\sqrt{n}  - 2\sqrt{n+1} = -2({\sqrt{n+1} - \sqrt{n}}) = -2(({\sqrt{n+1} - \sqrt{n}})\frac{\sqrt{n+1} + \sqrt{n}}{\sqrt{n+1} + \sqrt{n}}) = -2\frac{1}{\sqrt{n+1} + \sqrt{n}} > -2\frac{1}{2\sqrt{n+1}} = \frac{-1}{\sqrt{n+1}} = b_n $$ 
and we know that:
$$\sum_{n=1}^\infty b_n$$
doesnt converge cause 
$$\sum_{n=1}^\infty \frac{1}{\sqrt{n}}$$
doesnt converge
so from here my conclusion is that $\sum_{n=1}^\infty a_n$ doesnt converge
but i know the final answer is that it does converge 
so what am i doing wrong?

$$\begin{eqnarray*}\sum_{n=1}^{N}\left(2\sqrt{n}-\sqrt{n-1}-\sqrt{n+1}\right)&=&\color{red}{\sum_{n=1}^{N}\left(\sqrt{n}-\sqrt{n-1}\right)}+\color{blue}{\sum_{n=1}^{N}\left(\sqrt{n}-\sqrt{n+1}\right)}\\&=&\color{red}{\sqrt{N}}+\color{blue}{1-\sqrt{N+1}}\\&=&1-\frac{1}{\sqrt{N}+\sqrt{N+1}},\tag{1}\end{eqnarray*}$$
so, by letting $N\to +\infty$, we get that the original series converges towards $\color{green}{1}$.
