Find Fourier expansion of x² where x in (-2𝜋,2𝜋)

\[ f(x)=a_0 + \sum_{n=1}^{\infty} a_n \cos(\frac{n \pi x}{2 \pi}) + \sum_{n=1}^{\infty} b_n \sin(\frac{n \pi x}{2 \pi}) \]

\[ f(x)=a_0 + \sum_{n=1}^{\infty} a_n \cos(\frac{n x}{2}) \]

\[ a_0 = \frac{1}{4 \pi}\int_{-2\pi}^{2\pi} x^2 \, dx = \frac{2}{4 \pi}\int_{0}^{2\pi} x^2 \, dx = \frac{8\pi^{2}}{3} \]

\[ a_n = \frac{1}{L} \int_{-L}^{L} x^{2} \cos (\frac{n \pi x}{L}) dx = \frac{16}{n^{2}}(-1)^{n} \]

Find Fourier expansion of x² where x in (-2𝜋,2𝜋)