(Solved): Consider the function \( f \) defined on the interval \( [-2,2] \) as follows, \[ f(x)=\frac{1}{2} ...

Consider the function \( f \) defined on the interval \( [-2,2] \) as follows, \[ f(x)=\frac{1}{2} x . \] Denote by \( f_{F} \) the Fourier series expansion of \( f \) on \( [-2,2] \), \[ f_{F}(x)=\frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left[a_{n} \cos \left(\frac{n \pi x}{L}\right)+b_{n} \sin \left(\frac{n \pi x}{L}\right)\right] . \] Find the coefficients \( a_{0}, a_{n} \), and \( b_{n} \), with \( n \geqslant 1 \). \[ a_{0}= \] \[ a_{n}= \] \[ b_{n}= \]