Experiments with Fourier Series

Part 4: Fourier Approximations and the Series

1. Graph the function f given by



where f has been extended with period 2 pi.

2. Calculate the Fourier coefficients and display the Fourier approximations for f. For this example, you need exact expressions for the Fourier coefficients, not decimal approximations.
3. Study the approximations. Pay particular attention to the points at x = -pi and x = pi.
4. Assume that the Fourier series converges to f( x ) for all x such that -pi < x < pi. Show that the convergence at x = pi implies that

.

This last calculation should be exact -- numerical approximations are not good enough.