# Summation involving sine and cosine

1. Aug 31, 2011

### zje

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

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

I need to find $$\omega^2$$ and $$\frac{\partial\omega^2}{\partial K}$$, 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
$$\sin A \cos B = \frac{1}{2} [ \sin(A-B)+\sin(A+B)]$$
but I'm still worried that the $$\frac{1}{n}$$ 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. Sep 1, 2011

### lanedance

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

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

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