Q$5(25=10+10+5$ pts$.)$ Let $L^2(2)S_f(x)L^2(2)$;$(b)S_f(x)=f(x)L^2(2)S_f(-19)=(a)L=,T=2,=1.$ Then

$a_m=\frac{1}{}{\displaystyle \underset{-}{\overset{0}{}}}3xcos(mx)dx=\frac{3}{m}{\displaystyle \underset{-}{\overset{0}{}}}xdsm(mx)$

$=\frac{3}{m^2}(1-(-1{)}^m)=\{\begin{array}{ccc}0& ifm\text{e}\text{v}\text{e}\text{n}& \end{array}$

$\frac{6}{m^2}ifmis$ odd. $,m>0$

$a_0,=-\frac{3}{2},$

$b_m,=\frac{1}{}{\displaystyle \underset{-}{\overset{0}{}}}3xsin(mx)dx=\frac{-3}{m}{\displaystyle \underset{-5}{\overset{0}{}}}$xdcos$(mx)$

$,=\frac{3}{m}(-1{)}^{m+1},S_0,$

$S_f(x),=[-\frac{3}{4}+{\displaystyle \underset{n=1}{\overset{}{}}}\frac{6}{(2n-1{)}^2}cos((2n-1)x)+\frac{3(-1{)}^{n+1}}{n}sin(nx).$