Infinite Cosine Series

Geometry Level 5

cos(θ)5+2cos(2θ)25+3cos(3θ)125+4cos(4θ)625+5cos(5θ)3125+ \dfrac{\cos(\theta)}{5}+\dfrac{2\cos (2\theta)}{25}+\dfrac{3\cos (3\theta)}{125}+\dfrac{4\cos (4\theta)}{625}+\dfrac{5\cos (5\theta)}{3125}+\cdots

Let θ\theta be an acute angle with cosθ=710\cos\theta = \dfrac{7}{10}. If the value of the series above can be expressed as mn \dfrac mn, where mm and nn are coprime positive integers, find the value of m+nm+n.

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