In this mathlet I provide an interactive and visually engaging tool for exploring different types of convergence of sequences and series of functions. You can enter an arbitrary sequence or series of functions and choose ranges for x and y. The mathlet will plot one-by-one the consecutive terms or partial sums. In a separate box, you can enter a limit function as well. The limit function is allowed to be piecewise defined.
The mathlet contains many predefined examples of interesting sequences and series. These include the Maclaurin series for sin(x), cos(x), and other functions, the Fourier series for a sawtooth wave and a rectangular wave, and many sequences that illustrate interesting phenomena. These include a sequence of continuous functions converging to a limit that is not continuous, a sequence of functions whose integrals do not converge to the integral of the limit, sequences that are clearly uniformly convergent, and sequences that are convergent only pointwise.
Open Sequences and Series of Functions in a new window
The mathlet will run on any machine with a generic browser as long as it has Flash Player 6 or higher. The free and small (668KB) Player can be easily downloaded and installed from the Macromedia site -- click on the button at the right. Netscape 6 or higher usually comes with the Player ready to use.