Summation involving sine and cosine

  • Thread starter zje
  • Start date
  • #1
zje
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Homework Statement


[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.


Homework Equations


Not sure, the stuff above.


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:

Answers and Replies

  • #2
lanedance
Homework Helper
3,304
2
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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