Sum of Cosines

matlabber

1. The problem statement, all variables and given/known data
I try to simplify to get rid of sum
[tex] \sum_{k=0}^{n-1}cos(2 \pi fk)[/tex]

2. Relevant equations


3. The attempt at a solution

I discover I shall use euler equation to form:


[tex] \sum_{k=0}^{n-1}\frac{1}{2}(e^{2 \pi fki}+e^{-2 \pi fki})[/tex]

but how to sum exponentials?

vela

Aren't those geometric series?

matlabber

but do I include exp() when I do geometric series?

vela

You need to express the terms in the form Ar^k. Use whatever A and r allow you to do this.

matlabber

is it [tex]\frac{1-exp(2 \pi fi)^{t}}{1-exp(2 \pi fi)}[/tex]

vela

If by t you mean n, that would be twice the sum of the first term. You might find it a little simpler to start with cos x = Re[e^ix]. Then you only have one term to deal with and no 1/2's floating around.

matlabber

thank you very much!!