Can anybody help in simplifying the trigonometric equation under Question 2 in the attached document?
Thanks
Your expression is:
$\displaystyle S(x)=\sum_{n=1}^{\infty}a_n \sin \left(\frac{(n+1/2)x}{h}\right)$
where:
$\displaystyle
a_n=\frac{\cos(n\pi)\sqrt{\beta}}{\sqrt{3+\frac{n^ 4}{a^4}}}
$
First of all observe that $\displaystyle \cos(n\pi)=(-1)^n$ and that $\displaystyle \sqrt{\beta}$ is a constant and so can be taken outside the summation.
$\displaystyle
a_n=a^4 \sqrt{\beta}\frac{(-1)^n}{\sqrt{3a^4+n^4}}
$
$\displaystyle
S(x)=a^4 \sqrt{\beta}\ \sum_{n=1}^{\infty}\frac{(-1)^n}{\sqrt{3a^4+n^4}} \sin \left(\frac{(n+1/2)x}{h}\right)
$
Next thought is: can we turn this into a Fourier series (that we recognise)? I don't have the time right now but may return to this.
CB