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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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  • $\begingroup$ Welcome to Math Stack Exchange, Redingold! You're likely to get more answers if you discuss your thoughts on the question you have, or your attempts at solving it! This will help anyone who wants to help you by demonstrating your understanding of the question. Happy asking! $\endgroup$ – Rob Bland Nov 28 '15 at 23:54
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Outline

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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