Consider the $2\pi$-periodic odd function defined on $[0, \pi]$ by $f(θ) = θ(π − θ)$.
(a) Draw the graph of $f$.
(b) Compute the Fourier coefficients of $f$, and show that $f(θ)=\frac{8}{\pi} \sum_{k\text{ odd } \geq 1} \frac{\text{sin k}\theta}{k^{3}}$
Could anyone tell me how to find the coefficients? When I applied the fourier series formulas of $b_{n}=\frac{1}{L}⋅∫f(x)(\frac{\text{sin}nπx}{L})dx, n>0$,
I get the result $\sum_{n=1}^{\infty}(\frac{−2(−1)^{n}πsin(nx)}{n}−\frac{4(−1)^{n}cos(nx)}{n^{2}})−\frac{π^2}{3}$ which is different from the above.
Why am I getting the cosine terms even though the function is odd?