A sum of Cosines (Fouries series)

[phonic]

Dear All,

I wonder how to calculate the sum of the following Cosines:
<br /> \sum_{n=1}^\infty \cos(nx)<br />

Can anyone give a hint? Thanks a lot!

 

[micromass]

First, use the following formula:

\cos \alpha= \frac{e^{i\alpha}+e^{-i\alpha}}{2}.

Then apply formulas of geometric series.

 

[dextercioby]

A little simpler is to directly compute

\sum_{n=1}^{\infty} (\cos nx + i\sin nx)

and at the end separate the real and imaginary parts. Also gives you the sume of sines.