Prove $\frac{1}{2} + \cos(x) + \cos(2x) + \dots+ \cos(nx) = \frac{\sin(n+\frac{1}{2})x}{2\sin(\frac{1}{2}x)}$ for $x \neq 0, \pm 2\pi, \pm 4\pi,\dots$

Question

I know that this can be proven inductively. However, I can't get passed the trig. I am pretty sure trig identities can show that the expression above is true for $n=0$, and that if the expression holds for $n=k$ it holds for $n=k+1$. But alas, I am getting lost in a sea of trig. Hopefully someone can shed some light on this.

@peterwhy That would be odd, since this is an important inequality to derive some identities concerning the Dirichlet kernel. — Pedro, Oct 09 '13 at 22:33
@PedroTamaroff well, not exactly using a lot of Fourier series, but deriving using an idea from frequency domain is possible. — peterwhy, Oct 09 '13 at 22:36
@peterwhy (I wouldn't say your answer is using Fourier series, but it is a nice one. +1) — Pedro, Oct 09 '13 at 22:39
Related : http://math.stackexchange.com/questions/17966/how-can-we-sum-up-sin-and-cos-series-when-the-angles-are-in-arithmetic-pro — lab bhattacharjee, Oct 10 '13 at 05:29

peterwhy · Answer 1 · 2013-10-09T23:06:03.627

6

Hint: $$\frac{1}{2} + \sum_{k=1}^n \cos(kx) = \frac{1}{2}\sum_{k=-n}^n e^{ikx}$$

edited Oct 09 '13 at 23:06

answered Oct 09 '13 at 22:32

peterwhy

22,256

Pocho la pantera · Answer 2 · 2013-10-09T22:23:20.053

5

Hint: $$ 2\cos(kx)\,\sin(\frac{x}{2})=\sin\left(kx+\frac{x}{2}\right)- \sin\left(kx-\frac{x}{2}\right) $$

edited Oct 09 '13 at 22:23

answered Oct 09 '13 at 22:21

Pocho la pantera

2,436

Prove $\frac{1}{2} + \cos(x) + \cos(2x) + \dots+ \cos(nx) = \frac{\sin(n+\frac{1}{2})x}{2\sin(\frac{1}{2}x)}$ for $x \neq 0, \pm 2\pi, \pm 4\pi,\dots$

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