Sum over greatest integer

Question

Evaluate ${\lim_{n \to \infty} \frac{1}{n}\sum_{k=1}^{[\frac{n}{2}]} \cos \left(\frac{ kπ}{n}\right)}$

I think it is Riemann sum of some integral like $\int_{0}^{1} \cos (πx) \,dx ,$ but how to approach for the upper limit of summation?

Answer 1

This limit can be changed to an Integral$$L=\lim_{n \rightarrow \infty} \sum_{k=1}^{[n/2]} \cos \left(\frac{k \pi}{n}\right)= \int_{0}^{1/2} cos (\pi x) dx= \frac{1}{\pi}. $$