Suppose we evaluate the Fourier coefficients $a_n$ and $b_n$ from the function $x^2$ and then find the Fourier series. Whose Fourier series is it?
Is it the Fourier series of the function given by
$f(x)=x^2, x\in[-π,π]$
or,
$f(x)=x^2,x\in(-π,π]$, $f(x+2π)=f(x)$?
A Fourier series on $[-\pi,\pi]$ can either be viewed as approximating a function on $[-\pi,\pi]$ with no periodicity assumed, or else as a function on all of $\mathbb{R}$ with period $2\pi$, or else as a function on the unit circle. The choice is entirely up to you; they are equivalent. What do you want the domain of your function to be?
So if we compute the Fourier coefficients of a non-periodic function such as $x^2$, then according to our three choices, the resulting Fourier series can either be viewed as approximating the function $x^2$ restricted to $[-L,L]$, or it can be viewed as approximating the function
$$ x\mapsto\begin{cases} \dotsc\\ x^2 & x\in[-L,L]\\ (x-L)^2 & [L,2L]\\ (x-2L)^2 & [2L,3L]\\ \dots \end{cases} $$ on all of $\mathbb{R}$, or else it can be $e^{\pi i\theta/L}\mapsto\theta^2$ on the unit circle.
A Fourier series is a sum of periodic functions, so it is always periodic. It can never converge in any sense to a non-periodic function outside of the domain of periodicity. Instead it will converge to the piecewise function you get by forcing the function to be periodic by repeating the output from the fundamental domain.