Which methods different than the natural one can one devise to confirm that the limit is $\;2/\pi\;$? Good evening,
I have found this exercise (https://math.stackexchange.com/questions/633509/which-methods-different-than-the-natural-lim-n-to-infty-frac-cos1-cos)
What is the limit of:
$$\lim_{n\to\infty}\dfrac{|\cos{1}|+|\cos{2}|+|\cos{3}|+\cdots+|\cos{n}|}{n}$$
I am trying to solve it (without Probability, I have no knowledge on these subjects) but I didn't succeed, is it possible to solve it? If so, how to find the limit?
Thanks in advance.

Hint: If one replaces $n$ with $n\bmod {2\pi}$, the quotient tuens into something like a "distorted" Riemann sum. How distorted can it be? What does it mean of $n_1\bmod {2\pi}$ and $n_2\bmod{2\pi}$ are "unusually" close to each other?
