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Homework Help: Complex variables

  1. Sep 28, 2010 #1
    1. The problem statement, all variables and given/known data
    Let w = e^((2pi*i)/n). Show that 1+2w+3w^2+......+nw^(n-1) = n/(w-1)


    2. Relevant equations
    1+x+x^2+x^3+....+x^m = (1-x^(m+1))/(1-x) --> A
    1+x+x^2+x^3+.... = 1/(1-x) --> B

    3. The attempt at a solution
    First of all, multiply (w-1) on both sides, then we get
    w+2w^2+3w^3+...+(nw^n)-1-2w-3w^3-...-nw^(n-1) = n we simplify left side,
    -1-w-w^2-w^3-...-w^n+(nw^n) = n add -1 on both sides
    1+w+w^2+w^3+...+w^n-(nw^n) = -n
    1+w+w^2+w^3+...+w^n = (nw^n)-n for the left side, we know from A that
    (1-w^(n+1))/(1-w) = (nw^n)-n
    But I can't get left side and right side equal.
    Did I use the right method? Which part is wrong?
    Please tell me...
    Thank you
     
  2. jcsd
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