Infinite sum

Calculus Level 5

For b>1b>1 and cosx=a\cos x = a, find the value of :

r=0cos(rx)br\displaystyle \sum_{r=0}^{\infty} \frac{\cos(rx)}{b^r}.

If for b=32b = \frac{3}{2} and a=12 a = \frac{1}{2}, this value can be expressed as pq\frac{p}{q},where pp and qq are coprimes, then find pqpq.

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