Firstly, it is not that hard to notice that $1/(\sqrt{1} + \sqrt{3}) + 1/(\sqrt{3} + \sqrt{5}) + \cdots + 1/(\sqrt{2n-1} + \sqrt{2n+1}) = \frac{\sqrt{2n+1}- 1}{2\sqrt{n}}$
Hence, it is pretty easy to claculate limit, it is $\frac{\sqrt{2}}{2}$.
But what I am looking for is finding interdependence between $N$ and $\epsilon$ from definition:
$$\forall \epsilon>0\;\;\exists N \in \mathbb{N}:\;\forall n \geq N \implies \left|x_n - \frac{\sqrt{2}}{2}\right|<\epsilon$$
As an example, for sequence $y_n=\frac{1}{n}$, $N(\epsilon) = \lceil1/\epsilon\rceil$. So, for any positive $\epsilon$ we basically find interdependence such that we can find $N$ very quick.
But in my problem I struggle with finding such $N(\epsilon)$. All hints and solutions will be appreciated!
To start:
$$\begin{align}\left|x_n-\frac {\sqrt2}2\right|&=\left|\frac{\sqrt{2n+1}-1}{2\sqrt n}-\frac {\sqrt2}2\right|\\ &=\frac1{2\sqrt n}\left|\sqrt{2n+1}-1-\sqrt{2n}\right|\\ &=\frac1{2\sqrt n}\left|\frac 1{\sqrt{2n+1}+\sqrt{2n}}-1\right|\\ &=\frac1{2\sqrt n}\left(1-\frac 1{\sqrt{2n+1}+\sqrt{2n}}\right)\\ &<\frac1{2\sqrt n} \end{align}$$