Why do Fourier Series only work from $- π$ to $π$?

Question

Take Euler's famous example:

$$\dfrac{1}{2} x= \sin x-\dfrac{1}{2} \sin 2x+ \dfrac{1}{3} \sin 3x- \dfrac{1}{4} \sin 4x+\cdots $$

What is the reason this only works on $[-π,π]$?

Answer 1

Because you are not approximating the function $f(x)=x/2$, you are approximating the function $$f(x)=\begin{cases}x/2&x\in[-\pi,\pi]\\f\left(x-\lfloor\frac{x+\pi}{2\pi}\rfloor(2\pi)\right)&x\notin[-\pi,\pi]\end{cases}$$ which is periodic. So technically it does work for all $x\in\mathbb{R}$, just not for the function you were thinking of.