The Gibbs phenomenon is the fact that even when the Fourier series of a function $f$ converges at some point $t_{0}$ the convergence may be non-uniform in any neighborhood of $t_{0}$. We can find a sequence $t_{k}$ that converges to $t_{0}$ but with $lim_{n\rightarrow \infty} S_{n}(t_{n}) \neq f(t_{0})$. This happens whenever $f$ has a jump discontinuity at $t_{0}$.
Let $$ H(t) = \begin{cases} 1 & { 0<t< \pi}, \\ -1 & {-\pi< t < 0} \end{cases} $$ Compute $S_{n}(H)$ for $n \geq 1.$
Could anyone help me please?
You have first to consider your (odd) function as coinciding with a square wave on $(-\pi,\pi)$ and, thus, have a corresponding Fourier series identical to this square wave, i.e.,
$$\tfrac{4}{\pi}\sum_{k=0}^{\infty} \tfrac{1}{2k+1}\sin{((2k+1)t)}$$
Then, see the very clear document (http://web.mit.edu/jorloff/www/18.03-esg/notes/gibbs-phenom.pdf)