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Verify infinite series

  1. Apr 16, 2009 #1
    1. The problem statement, all variables and given/known data

    Show that for all integers n [tex]\geq[/tex] 1,
    cos(2x) + cos(4x) + ... + cos(2nx) = [tex]\frac{1}{2}[/tex] ([tex]\frac{sin((2n+1)x)}{sin(x)}[/tex]-1)

    Use this to verify that
    [tex]\sum_{n=1}^{\infty}(\int_{0}^{\pi}[/tex] x([tex]\pi[/tex]-x)cos(2nx)dx) =

    [tex]\frac{-1}{2}\int_{0}^{\pi}[/tex] x([tex]\pi[/tex]-x)dx)

    2. Relevant equations

    3. The attempt at a solution
    I proved the first part of this problem using induction, however I don't see how I can use that to verify the second part. Maybe I can bring the summation into the integral and get the sum of cos(2nx), but I still don't see how that would give me what I need to prove. Any suggestions?
  2. jcsd
  3. Apr 17, 2009 #2


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    Science Advisor
    Homework Helper

    If you interchange the sum with the integral, you can substitute the identity you have proved.
    Then the second part (with the - 1) will give you the answer you want, you will just need to prove that
    [tex]\lim_{n \to \infty} \frac12 \int_0^\pi x (\pi - x) \frac{\sin((2n+1)x)}{\sin(x)}\, dx = 0[/tex]
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