Chapter 12.6, Problem 19E

To determine

To show: If $f$ is an even function on interval $\left(-1,1\right)$, then $f(x)={\displaystyle \sum _{n=0}^{\infty }{c}_{2n}{P}_{2n}(x)}$ and $c_{2n}=(4n+1){\displaystyle \underset{0}{\overset{1}{\int }}f(x)}{P}_{2n}(x)dx$.