Chapter 10.2, Problem 21P

a.

To determine

To graph: The function defined by $f\left(x\right)=\frac{x^2}{2}$ in the interval $-2\le x<2$ with $f\left(x+4\right)=f\left(x\right)$ for three different periods.

b.

To determine

The Fourier series for the function $f\left(x\right)=\frac{x^2}{2}$.

c.

To determine

To plot: The partial sums corresponding to the Fourier series $f\left(x\right)=\frac{2}{3}+{\displaystyle \sum _{n=1}^{\infty }\frac{8{\left(-1\right)}^n}{n^2{\pi }^2}\cos \frac{n\pi x}{2}}$ taking the partial sum $s_m\left(x\right)$ along the y-axis for $m=5$, $m=10$ and $m=20$.

d.

To determine

To describe: The convergence of the Fourier series $f\left(x\right)=\frac{2}{3}+\frac{8}{{\pi }^2}{\displaystyle \sum _{n=1}^{\infty }\frac{{\left(-1\right)}^n}{n^2}\cos \frac{n\pi x}{2}}$.