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Tricky sum

  1. Jan 22, 2008 #1

    nicksauce

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    1. The problem statement, all variables and given/known data
    Evaluate the sum
    [tex]\sum_{n=0}^N\frac{\cos{n\theta}}{\sin^n{\theta}}[/tex]

    2. Relevant equations



    3. The attempt at a solution
    In class we evaluated [tex]\sum_{n=0}^N\cos{n\theta}[/tex] and [tex]\sum_{n=0}^N\sin{n\theta}[/tex], by expanding them as the real and imaginary parts of a geometric series. However, I can't quite seem to figure out to use that for this question. Could someone give me a bump in the right direction?
     
  2. jcsd
  3. Jan 22, 2008 #2
    Maybe De Moivre's Theorem is useful here? Not sure if that's what you meant by expanding as real and imaginary parts.

    [cos(theta) + i*sin(theta)]^n = cos(n*theta) + i*sin(n*theta)
     
  4. Jan 22, 2008 #3

    nicksauce

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    For example, we used

    [tex]\sum_{n=0}^{N}\cos{n\theta} = Re(\sum_{n=0}^{N}z^n)[/tex]

    And then use the analytic formula for the RHS.
     
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