For the fourier series of $\sin(x)$ for the given domain, I got
$$-\,\frac{8}{\pi}\sum^{\infty}_{n = 1}\left(-1\right)^{n} \,{n \over 4n^{2} - 1}\,\sin\left(2nx\right)$$ Now after using this result to calculate the first four plots ( approximations for part $\left(\mathrm{b}\right)$ ), as I increase the n value in the fourier approximation, I'm getting results that look less and less like $\sin\left(x\right)$. Shouldn't the fourier series of $\sin\left(x\right)$ ultimately converge to $\sin\left(x\right)$ ?. Does this mean I did something wrong ?.