I have two infinite sums, and I can't figure out how to evaluate them.

The first is $\sum_{n = 0}^{\infty }2^{-n}\cos(n\theta)$ and the second is $\sum_{n = 0}^{\infty }2^{-n}\sin(n\theta)$.

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Let $S_1 = \sum_{n = 0}^{\infty }2^{-n}\cos(n\theta)$ and $S_2 = \sum_{n = 0}^{\infty }2^{-n}\sin(n\theta)$.

Consider $S = S_1 + iS_2$

Then $S = \sum_{n = 0}^{\infty }2^{-n}e^{in\theta}$.

This is a geometric series and can be summed easily. Simplify the expression and then take real and imaginary parts to get your answers.

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