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.

Using the Euler-Maclaurin formula we can get more detailed asymptotics, e.g.:
$$ u_n =  \left( \sqrt {5}-\sqrt {2} \right) n
+ \frac{\sqrt{5}}{5} + \frac{\sqrt{2}}{4}
+\left(\frac{\sqrt{5}}{300}- \frac{\sqrt{2}}{48}  \right) n^{-1} +
\left(-\frac{\sqrt{5}}{10000} + \frac{\sqrt{2}}{1280} \right) n^{-3}
+O \left( {n}^{
-5} \right) $$
