Chapter 7.8, Problem 15P

Sketch (or computer plot) each of the following functions on the interval $(-1,1)$ and expand it in a complex exponential series and in a sine-cosine series.

(a) $f(x)=x,-1<x<1$.

Answer: $f(x)=\frac{2}{\pi }{\displaystyle \underset{1}{\overset{\infty }{\sum }}{(-1)}^{n+1}}\frac{\sin n\pi x}{n}$.

(b)

Answer: $f(x)=\frac{8}{{\pi }^2}{\displaystyle \underset{\text{odd }n=1}{\overset{\infty }{\sum }}\frac{\cos n\pi x}{n^2}}.$

(c)