Show convergence and calculate the limit: $\lim \limits_{n \to \infty}\sum \limits_{k=n}^{2n}\frac{k}{k^2+n^2}$

Question

I guess it has something to do with Riemann sums but this is new for me.

$\displaystyle\lim \limits_{n \to \infty}\sum \limits_{k=n}^{2n}\frac{k}{k^2+n^2}$

How do I start?

Answer 1

Like this: $$ \sum_{k=n}^{2n}\frac{k}{k^2+n^2}=\frac1n\sum_{i=0}^{n}\frac{1+\frac{i}n}{\left(1+\frac{i}n\right)^2+1}. $$

Answer 2

$$S_n = \sum_{k=n}^{2n} \dfrac{k}{k^2+n^2} = \sum_{k=n}^{2n} \dfrac{k/n}{(k/n)^2+1} \dfrac1n$$ Let $x_{k-n+1} = k/n$, we then have $$S_n = \sum_{k=1}^n \dfrac{x_k}{1+x_k^2} \Delta x_k$$ You should be able to relate the above sum to $$\int_1^2 \dfrac{x}{1+x^2} dx$$