Question

Fourier Series Help

http://i.imgur.com/5JSvN8P.png

(I'll post my working in another post)

Answer 1

So I know the Fourier sine series expansion is define as:\[f(x)=\sum_{n=1}^{\infty}b_nsin(nx)\]
I worked out b_n like so:\[b_n=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)sin(nx)dx\]\[b_n=\frac{2}{\pi}\int_{0}^{\pi}sin(nx)dx\]\[b_n=\frac{2}{\pi}\left[ -\frac{1}{n}\cos(nx) \right]_{0}^{\pi}\]\[b_n=\frac{2(1-cos(n\pi))}{n\pi}\]Then I put this back into f(x) to get
\[f(x)=\frac{2}{\pi}\sum_{n=1}^{\infty}\frac{(1-cos(n\pi))sin(nx)}{n}\]
Which isn't correct. Can anyone see where I'm going wrong?