Why can Fourier series be used on a non-repetitive function?
Fourier series can be calculated for any signal in close section (from $T_0$ to $T_1$). In fact, I was taught that this is how Fourier series is defined.

But Fourier series is made up of ($T_0 - T_1$) cyclic functions, therefore, if you go beyond the original section you will just get repetitions as you displayed.