Sequence: $u_n=\sum_{k=n}^{2n}\frac{k}{\sqrt{n^2+k^2}}$ 
Study the following sequence of numbers:
  $$\forall n\in\mathbb{N}, u_n=\sum_{k=n}^{2n}\frac{k}{\sqrt{n^2+k^2}}$$

I tried to calculate $u_{n+1}-u_n$, but I couldn't simplify the expression.
Plotting the sequence shows arithmetic (or seems to be an) progression.

We have that
$$\sum_{k=n}^{2n}\frac{k}{\sqrt{n^2+k^2}} \ge n\cdot \frac{n}{\sqrt{2n^2}}=\frac{n}{\sqrt 2}$$
form which we conclude that $u_n \to \infty$, we can also obtain that
$$\frac{n}{\sqrt{2}}\le \sum_{k=n}^{2n}\frac{k}{\sqrt{n^2+k^2}} \le \frac{2n}{\sqrt{5}}$$
and by Riemann sum since
$$\lim_{n\to \infty }\frac1n \sum_{k=n}^{2n}\frac{k}{\sqrt{n^2+k^2}}= \lim_{n\to \infty }\frac1n\sum_{k=n}^{2n}\frac{\frac kn}{\sqrt{1+\frac{k^2}{n^2}}}=\int_1^2 \frac{x}{\sqrt{1+x^2}}dx=\sqrt 5 - \sqrt 2$$
we obtain
$$\sum_{k=n}^{2n}\frac{k}{\sqrt{n^2+k^2}}\sim (\sqrt 5 - \sqrt 2)n$$
