Sum of evenly spaced cosines

Question

I am trying to evaluate $\lim \limits_{n \to \infty} {\cos2 + \cos4 + \cos6 + \dots + \cos{2n} \over n}.$ I got a hint that this might be the trigonometric form I'm aiming for and that $z = \cos\theta + i\sin\theta$, $z^n = \cos{n\theta} + i\sin{n\theta}$ might be useful here, however, I can't see the connection. Intuition suggests the limit is $0$.

Answer 1

Let $z=e^{2i}$ and

$$S_n=z+z^2+z^3+...z^n =z\frac{z^n-1}{z-1}$$

$$=ze^{i(n-1)}\frac{\sin(n)}{\sin(1)}.$$

$Re(S_n)=$ $\cos(n+1)\frac{\sin(n)}{\sin(1)}.$

and

$|Re(S_n)|\leq \frac{1}{\sin(1)}$.

thus $$\lim_{n\to+\infty}\frac{\displaystyle{\sum_{k=1}^n\cos(2k)}}{n}=\lim_{n\to +\infty}\frac{Re(S_n)}{n}=0.$$