Tricky Trigonometric Summation

Calculus Level 4

The sum n=0cos(nθ)2n\sum_{n=0}^{\infty} \frac{\cos{(n \theta)}}{2^{n}} where cosθ=15\cos{\theta} = \frac{1}{5}, can be expressed in the form αβ\frac{\alpha}{\beta} where α\alpha and β\beta are coprime positive integers. Find α+β\alpha + \beta.

Credit to MAΘ\Theta 1991

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