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How to simply this sum? $\cos\theta+\cos\frac{\theta}{2}+\cos\frac{\theta}{2^2}+\dots+\cos\frac{\theta}{2^n}$. What about the $\sin$ function case?

I tried to use double angle formula for cosine. But I couldn't even simply the sum of the first two terms. I think this kind of formula can be very tricky. If you give me the final answer, then I can prove it by induction. It is hard to guess the final answer.

I think this is a kind of math Olympics problem.

Tony B
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  • What have you done till now –  Oct 28 '15 at 13:55
  • @Rememberme I tried to use double angle formula for cos. But I couldn't even simply the sum of the first two terms. I think this kind of formula can be very tricky. If you give me the final answer, then I can prove it by induction. It is hard to guess the final answer. – Tony B Oct 28 '15 at 14:47
  • Related http://math.stackexchange.com/questions/910670/naive-proof-that-sum-n-1n-1-cos2-pi-fracnn-1 – cgiovanardi May 13 '16 at 12:25
  • @cgiovanardi Thanks. However, the angles in that problem form an arithmetic progression, angles in my problem form a geometric series. – Tony B Jul 11 '16 at 22:39

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