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Summation involving sine and cosine

  1. Aug 31, 2011 #1

    zje

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    1. The problem statement, all variables and given/known data
    [tex]\omega^2=(2/M)\sum_{n>0}\frac{A\sin(nk_0a)}{na}(1-\cos(nKa))[/tex]

    A, a, and k_0 are constants, n is an integer.

    I need to find [tex]\omega^2[/tex] and [tex]\frac{\partial\omega^2}{\partial K}[/tex], but I have no idea where to start.


    2. Relevant equations
    Not sure, the stuff above.


    3. The attempt at a solution
    I haven't done something like this in a while. I think I might be able to exploit
    [tex] \sin A \cos B = \frac{1}{2} [ \sin(A-B)+\sin(A+B)][/tex]
    but I'm still worried that the [tex]\frac{1}{n}[/tex] term is divergent...

    Any ideas on where to start? I'm also considering a complex approach with Euler's formula, but I'm not sure how far I can go with this.


    Many thanks!
     
    Last edited: Sep 1, 2011
  2. jcsd
  3. Sep 1, 2011 #2

    lanedance

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    Homework Helper

    I would just differentiate directly, considering omega squared as a function of K

    [tex]
    f(K) = \omega^2(K)=(2/M)\sum_{n>0}\frac{A\sin(nk_0a)}{na}(1-\cos(nKa))
    [/tex]

    [tex]
    \frac{d}{dK}f(K) = \frac{d}{dK}\omega^2(K)=\frac{d}{dK}(2/M)\sum_{n>0}\frac{A\sin(nk_0a)}{na}(1-\cos(nKa))
    [/tex]
    [tex]

    =(2/M)\sum_{n>0}\frac{A\sin(nk_0a)}{na}(\frac{d}{dK}\cos(nKa))
    [/tex]
     
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