Chapter 7.11, Problem 2P

Prove that if $f(x)={\displaystyle \underset{i=\infty }{\overset{\infty }{\sum }}{c}_n}{e}^{inx},$ then the average value of ${[f(x)]}^2$ is ${\displaystyle \underset{-\infty }{\overset{\infty }{\sum }}{c}_n}{c}_{-n}$.

Show by Problem 7.12 that for real $f(x)$ this becomes (11.5).