Hard limit of sine function How can I compute this limit : $$\lim_{n \to \infty } {1 \over n}\sum\limits_{k = 1}^n {\left\lvert \sin k\right\rvert} $$
Thank you.

By Weyl's equidistribution theorem the sequence $\{e^{in}\}_{n\geq 0}$ is dense in the unit circle and much more: it is equidistributed. In particular, since $\sin(k)=\text{Im}\,e^{ik}$, the limit $\lim_{n\to +\infty}\frac{1}{n}\sum_{k=1}^{n}\left|\sin(k)\right|$ is the average value of the function $\left|\sin(x)\right|$, i.e.
$$ \frac{1}{\pi}\int_{0}^{\pi}\sin(x)\,dx = \color{red}{\frac{2}{\pi}}.$$
