I'm asked to obtain a Fourier series for the function
$$f(x) = x, [0, \pi]$$
in a fashion such that the period of the expansion is $4\pi$ and the expansion must be in sines with an $n$ such that $n = 2k -1$, as in only $b_{2k - 1}$ coefficients will appear.
I was given the result that
$$b_{2k-1} = \dfrac{2}{\pi}\int_{0}^{\pi}f(x) \cdot \sin(\dfrac{n\pi x}{2\pi})\,dx$$ but I'm having a bit of trouble.
Thanks for any input.