Chapter 11.5, Problem 20E

To determine

To show: If $f$ is an odd function on interval $\left(-1,1\right)$, then $f(x)={\displaystyle \sum _{n=0}^{\infty }{c}_{2n+1}{P}_{2n+1}(x)}$ and $c_{2n+1}=(4n+3){\displaystyle \underset{0}{\overset{1}{\int }}f(x)}{P}_{2n+1}(x)dx$.