Chapter 10.3, Problem 28E

a. Show that the function $f\left(x\right)=x^2$ has the Fourier series, on $-\pi <x<\pi$,

$f\left(x\right)\sim \frac{{\pi }^2}{3}+4{\displaystyle \underset{n=1}{\overset{\infty }{\sum }}\frac{{\left(-1\right)}^n}{n^2}}\cos nx$.

b. Use the result of part (a) and Theorem 2 to show that

${\displaystyle \underset{n=1}{\overset{\infty }{\sum }}\frac{{\left(-1\right)}^{n+1}}{n^2}=}\frac{{\pi }^2}{12}$.

c. Use the result of part (a) and Theorem 2 to show that

${\displaystyle \underset{n=1}{\overset{\infty }{\sum }}\frac{1}{n^2}=}\frac{{\pi }^2}{6}$.