Chapter 10.2, Problem 19P

a.

To determine

To graph: The function defined by $f\left(x\right)=\left\{\begin{array}{l}-1,-\text{2}\le x<0\\ 1,\text{ 0}\le x<2\end{array}\right\}$ with $f\left(x+4\right)=f\left(x\right)$ for three different periods.

b.

To determine

The Fourier series for the function $f\left(x\right)=\left\{\begin{array}{l}-1,-\text{2}\le x<0\\ 1,\text{ 0}\le x<2\end{array}\right\}$.

c.

To determine

To plot: The partial sums corresponding to the Fourier series $f\left(x\right)=\frac{4}{\pi }{\displaystyle \sum _{n=1}^{\infty }\frac{\sin \frac{\left(2n-1\right)\pi x}{2}}{2n-1}}$ 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 $f\left(x\right)=\frac{4}{\pi }{\displaystyle \sum _{n=1}^{\infty }\frac{\sin \frac{\left(2n-1\right)\pi x}{2}}{2n-1}}$.