If $p$ is a prime then what is the value of the series $$\cos\frac{2\pi}{p}+\cos\frac{4\pi}{p}+\cos\frac{6\pi}{p}+\dots +\cos\frac{(p - 1)\pi}{p}$$ In general what is the value of the following series $$\cos\frac{k\pi}{p}+\cos\frac{2k\pi}{p}+\cos\frac{3k\pi}{p}+\dots +\cos\frac{2\frac{(p - 1)}{2}k\pi}{p}$$
Asked
Active
Viewed 81 times
1
1 Answers
3
In general, if $n$ is an odd positive integer,
$$\sum_{k=0}^{(n-1)/2}\cos(2\pi k/n)=\Re \sum_{k=0}^{(n-1)/2}\exp(2\pi i k/n) =\Re\frac{e^{2\pi i (n-1)/2n}-1}{e^{2\pi i/n}-1} =\Re\frac{-e^{\pi i/n}-1}{e^{2\pi i/n}-1} =\frac12$$
so
$$\sum_{k=1}^{n-1}\cos(2\pi k/n)=\frac12-1=\frac12.$$
The second sum can be handled similarly.
user246336
- 3,579
-
beautiful use of power sum – mathreadler Jun 14 '18 at 11:54
-
Thanks @iqcd for your answer. But limit of k is upto (p-1)/2. I think you have considered k upto (p-1). – idiot Jun 14 '18 at 13:10
-
@idiot You're right, my bad. Now it should be fixed – user246336 Jun 14 '18 at 13:36
-
Thanks. Got it. – idiot Jun 14 '18 at 13:54