How can each and every periodic signal be represented as Fourier Series

Question

So this seems to me quite surprising of when I see discontinuous although some continuous functions. As of how to represent them in Fourier Series. Can someone provide answers suggesting the examples

Answer 1

By "representing", you have to address what you want. Let $S_n(f)$ denote the first $n$ finite Fourier sum for a integrable function $f$. If $f\in L^p$ with $1<p<\infty$, we can show that $\lim_{n\to\infty}\|S_n(f)-f\|_{L^p}=0$ and for almost every $x$, $\lim_{n\to\infty}S_n(f)(x)=f(x)$. The pointwise convergence result is true even when $p=\infty$. Study of this problem belongs the branch of harmonic analysis/Fourier analysis.