Chapter 7.5, Problem 1P

In each of the following problems you are given a function on the interval $-\pi <x<\pi$.

Sketch several periods of the corresponding periodic function of period $2\pi .$ Expand the periodic function in a sine-cosine Fourier series.

In this case the sketch is:

Your answer for the series is: $f(x)=\frac{1}{2}-\frac{2}{\pi }\left(\frac{\sin x}{1}+\frac{\sin 3x}{3}+\frac{\sin 5x}{5}\cdots \right)$.

Can you use the ideas of Example 2 to find this result without computation?