How to find the minimum value of cos(2x)+cos(4x)+cos(6x)+cos(8x)+...+cos(20x) I want to find the minimum value of the series cos(2x)+cos(4x)+cos(6x)+cos(8x)+...+cos(2nx). x could be 2pit. Anyone can share a method of how to determine the minimum value of the series?

Comment:
Let $t=2x$ , you get:
$s=\cos (t)+\cos(2t)+\cos(3t)+\cdot\cdot\cdot +\cos(10t)$
Now use this formula:
$$s=\cos (t)+\cos(2t)+\cos(3t)+\cdot\cdot\cdot +\cos(nt)=\frac{\sin \frac n2 t\cos\frac {n+1}2 t}{\sin (\frac t 2)}$$
So you have to find minimum of this function:
$$y=\frac{\sin(10x)\cos(11x)}{\sin(x)}$$
By plotting in Wolfram minimum of y is about -2.8.
