Tricky Trigonometric Summation

Calculus Level 4

The sum \[\sum_{n=0}^{\infty} \frac{\cos{(n \theta)}}{2^{n}}\] where \(\cos{\theta} = \frac{1}{5}\), can be expressed in the form \(\frac{\alpha}{\beta}\) where \(\alpha\) and \(\beta\) are coprime positive integers. Find \(\alpha + \beta\).

Credit to MA\(\Theta\) 1991