Chapter 5.4, Problem 50PE

In this section, we used the product-to-sum formulas to write sums and differences of trigonometric functions. Another type of expression involving the sum or difference of sine and cosine terms is a Fourier series named after Jean-Baptiste Joseph Fourier (1768-1830). A Fourier series is an expression of the form $A_0+\left(A_1\cos x+B_1\sin x\right)+\left(A_2\cos 2x+B_2\sin 2x\right)+\left(A_3\cos 3x+B_3\sin 3x\right)+\mathrm{...}$ where $A_1,A_2,A_3,\mathrm{...}$ and $B_1,B_2,B_3,\mathrm{...}$ , are constants. Fourier proved that any continuous function can be represented by a Fourier series. In Exercises 49-50, we use Fourier polynomials (a finite number of terms from a Fourier series) to model a "saw-tooth" wave and a "square" wave.

A "square" wave is a periodic wave that alternates between two fixed values with equal time spent at each value and with negligible transition time between them. Square waves have practical uses in electronics and music, and because of their rectangular pattern, they are used in timing devices to synchronize circuits. Given,

$f\left(x\right)=\frac{4}{\pi }\left(\frac{1}{1}\sin \pi x+\frac{1}{3}\sin 3\pi x+\frac{1}{5}\sin 5\pi x+\cdot \cdot \cdot \right)$

a. Graph the first three terms of the Fourier series on the window $\left[-4,4,1\right]$ by $\left[-2,2,1\right]$ .

b. Graph the first five terms of the Fourier series on the window $\left[-4,4,1\right]$ by $\left[-2,2,1\right]$ .