Excuse my bad english.
Problem
Determine the real Fourier series for $f(t) = t^2$ with $t \in [0,1[$ converging as fast as possible.
My approach
Firstly I periodically expanded so that we get slide mirror symmetry. The function I ended up with is given by $$g\left(t\right)=\begin{cases} \left(t-0\right)^2&-1\le t\le \:1\\ -\left[\left(t-2\right)^2-1\right]&1\le \:t\le\:\:3\\ \end{cases}$$
I also graphed the piecewise function on desmos piecewise function $g(t)$
Secondly I started calculating the coëfficiënts $A_0$, $A_{2n}$, $B_{2n}$, $A_{2n-1}$, and $B_{2n-1}$ since there's a slide mirror symmetry we get $A_0=A_{2n}=B_{2n}=0$ and $$A_{2n-1}=\frac{4}{T}\int _0^{\frac{T}{2}}g\left(t\right)\cos\left(n\omega t\right)dt$$ and $$B_{2n-1}=\frac{4}{T}\int _0^{\frac{T}{2}}g\left(t\right)\sin\left(n\omega t\right)dt$$
My questions
$1$. Is this the best or fastest way to solve this problem or are the other (better) ways?
$2$. Have I approached this problem correctly?