Trying to compute this integral for my $b_n$'s i a Fourier series exercise I came out with this, eventually: $$b_n=\frac{8}{\pi n^3}$$ for odd integers and $$b_n=0$$ for even integers [Which is correct].
Now I want to write the series: $$f(x)=\sum_{n=1}^{\infty}b_n\sin\left ( \frac{n\pi x}{L} \right )$$ where $L=\pi$. So I think that substitute $n\to 2m+1$ would be correct, but actually the correct answer in my book is this: $$f(x)=\sum_{m=1}^{\infty}\frac{8}{\pi (2m-1)^3}\sin\left (2m-1 \right )x$$ Could any one explain why the correct iteration is $n\to 2m-1$?