Tricky Trigonometric Summation

Calculus Level 5

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