Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question

1 Answer 1

up vote 10 down vote accepted

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 $$2^n\cos^{n}\left(\frac{\theta}2\right)\cos\left(\frac{n\theta}2\right)$$

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.