Show the roots of unity add up to zero.

[zheng89120]

Homework Statement

Prove that $\Sigma$$^{n}_{k=1}$ w_k = 0

and there has to be at least two phasors/exponentials

Homework Equations

complex analysis

The Attempt at a Solution

I tried writing out the sigma on the first line.

Then I tried writing the same thing with n+1 on the second line.

Then I tried to divide the first exponential of the n+1 line by the first exponential of the first line, called A.

Then I divided the bottom by A, which should factor to get the first line as a factor, but could still not get the first line from the n+1, second line.

 

[Mark44]

Homework Statement

Prove that $\Sigma$$^{n}_{k=1}$ w_k = 0

and there has to be at least two phasors/exponentials

Homework Equations

complex analysis

The Attempt at a Solution

I tried writing out the sigma on the first line.

Then I tried writing the same thing with n+1 on the second line.

Then I tried to divide the first exponential of the n+1 line by the first exponential of the first line, called A.

Then I divided the bottom by A, which should factor to get the first line as a factor, but could still not get the first line from the n+1, second line.

It seems that you are trying to do a proof by induction, but you didn't state that very clearly. At least, that's what I think you were doing.

One thing that you are forgetting is that the things you are adding are roots of unity. In your original summation, (w_k)ⁿ = 1, for 1 <= k <= n.