$(2)$ Obtain the Fourier series for the function

${\displaystyle \underset{n=1}{\overset{}{}}}\frac{1}{n^2}=\frac{{}^2}{6}f(x)=\frac{1}{4}(-x{)}^2,0$

and deduce that

${\displaystyle \underset{n=1}{\overset{}{}}}\frac{1}{n^2}=\frac{{}^2}{6}$