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Show the roots of unity add up to zero.

  1. Sep 20, 2011 #1
    1. The problem statement, all variables and given/known data

    Prove that [itex]\Sigma[/itex][itex]^{n}_{k=1}[/itex] wk = 0

    and there has to be at least two phasors/exponentials

    2. Relevant equations

    complex analysis

    3. 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.
     
  2. jcsd
  3. Sep 20, 2011 #2

    Mark44

    Staff: Mentor

    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, (wk)n = 1, for 1 <= k <= n.
     
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