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De Moivre's Theorem and Power Series

  1. Jan 25, 2015 #1
    1. The problem statement, all variables and given/known data
    Hi I'm stuck with the following question:

    Use de Moivre's Theorem and your knowledge of power series to show:

    1/1(1/2^1)cos(θ)+1/2(1/2^2)cos(2θ)+1/3(1/2^3)cos(3θ)+ ... = log(2)-1/2*log(5-4cos(θ))


    2. Relevant equations


    3. The attempt at a solution
    I have already established the series to be (1/2)(∑((eiθ/2)^n/n) + ∑((e-iθ/2)n)/n) and evaluated the two series as a function of a natural logarithm ∑(x^n/n). But I'm not sure where to go from here, any help is much appreciated thanks.
     
  2. jcsd
  3. Jan 25, 2015 #2
    If you're stuck, try working it from the other direction. Start with ##log(2)- (1/2)log(5-4cos\theta)##.
     
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