$\displaystyle a_0\ =a_n\ =0 \\[2ex] \displaystyle and\ l=3, \\[2ex] \displaystyle b_n\ =\frac{2}{l}\int_0^lf\left(x\right)\sin\left(\frac{n\pi{}x}{3}\right)dx \\[2ex] \displaystyle =\frac{2}{3}\int_0^3x^2\sin\left(\frac{n\pi{}x}{3}\right)dx\ \\[2ex] \displaystyle \ \ \ \ =\frac{2}{3}{\left[x^2-\cos\left(\frac{n\pi{}x}{3}\right)\frac{3}{n\pi{}}-2x-\sin\left(\frac{n\pi{}x}{3}\right)\bullet{}\frac{3^2}{n^2{\pi{}}^2}+2\cos\left(\frac{n\pi{}x}{3}\right)\bullet{}\frac{3^3}{n^3{\pi{}}^3}\right]}_0^3\ \ \\[2ex] \displaystyle \ \ \ \ =\frac{2}{3}\left\{\left[3^2-{(-1)}^n\bullet{}\frac{3}{n\pi{}}-0+2{(-1)}^n\bullet{}\frac{3^3}{n^3{\pi{}}^3}\right]-\left[0-0+2\bullet{}1\bullet{}\frac{3^3}{n^3{\pi{}}^3}\right]\right\} \\[2ex] \displaystyle \ \ \ \ =\frac{2}{3}\bullet{}\frac{3^3}{\pi{}}\left\{-\frac{{(-1)}^n}{n}+\frac{2}{n^3{\pi{}}^2}\left[{\left(-1\right)}^n-1\right]\right\}\sin\left(\frac{n\pi{}x}{3}\right) \\[2ex] $

$\displaystyle \therefore{}x^2=\frac{18}{\pi{}}\left[\left(\frac{1}{1}-\frac{4}{1^3{\pi{}}^2}\right)\sin\left(\frac{1\pi{}x}{3}\right)-\frac{1}{2}\bullet{}\sin\left(\frac{2\pi{}x}{3}\right)+\left(\frac{1}{3}-\frac{4}{3^2{\pi{}}^2}\right)\bullet{}\sin\left(\frac{3\pi{}x}{3}\right)+…\right] \\[2ex]$