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The sum $$ 1 + {n \choose 1}\cos \theta + {n \choose 2}\cos 2\theta + \cdots+ {n \choose n}\cos n\theta $$ is?

I try to write this as the real part of $(1 + \cos \theta + i\sin \theta)^n$ but then I'm stuck.

Tunk-Fey
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Anamaki
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  • Related : http://math.stackexchange.com/questions/500325/displaystyle-sum-k-0n-frac-cosk-x-coskx, http://math.stackexchange.com/questions/364631/summing-sum-k-1n-k-cosk-theta-and-sum-k-1n-k-sink-theta and http://math.stackexchange.com/questions/769794/difficult-infinite-trigonometric-series – lab bhattacharjee May 09 '14 at 18:36

1 Answers1

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The given sum is the real part of

$$\sum_{k=0}^n{n\choose k}e^{ik\theta}=(1+e^{i\theta})^n=e^{in\theta/2}\left(2\cos\left(\frac{\theta}2\right)\right)^n$$ so the desired sum is $$2^n\cos^{n}\left(\frac{\theta}2\right)\cos\left(\frac{n\theta}2\right)$$