Summing cosines with Euler's formula

[matlabber]

Homework Statement

I try to simplify to get rid of sum
$$\sum_{k=0}^{n-1}cos(2 \pi fk)$$

Homework Equations

The Attempt at a Solution

I discover I shall use euler equation to form:$$\sum_{k=0}^{n-1}\frac{1}{2}(e^{2 \pi fki}+e^{-2 \pi fki})$$

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 $$\frac{1-exp(2 \pi fi)^{t}}{1-exp(2 \pi fi)}$$

 

[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!