complex numbers - Summation of $frac{cos n theta}{2^n}$

I would like to compute the following sum:

$$\sum_{n=0}^{\infty} \frac{\cos n\theta}{2^n}$$

I know that it involves using complex numbers, although I'm not sure how exactly I'm supposed to do so. I tried using the fact that $\cos \theta = {e^{i\theta} + e^{-i\theta}\over 2}$. I'm not sure how to proceed from there though. A hint would be appreciated.

Answer

Consider the series

$$S=\sum_{n=0}^{\infty}\left(\frac{e^{i\theta}}{2}\right)^n.$$
This is a geometric series whose sum is
$$S=\frac{2}{2-e^{i\theta}}.$$

Now the real part of $S$ is the sum you are looking for.