# Sixth laboratory - Fall 2002

## Warning

The section of the text most relevant to this lab is 11.2.

### Fourier series convergence

In this section of the lab you will investigate the convergence of the Fourier series of two functions: a triangular wave and a square wave. The Java applet tool below will graph any partial sum sn of either of these series. You specify which function and which n to use, and the intervals for the horizontal and vertical axes (x and y, then click "Plot function" to see the graph of sn, or "Plot error" to see f(x) - sn. If you click on the graph, the coordinates of the point where you clicked are shown near the top left of the graph.

Note that to get accurate results, you'll want to "zoom in" on certain points by choosing appropriate intervals for x and y.

#### Questions

Let en(x) = f(x) - sn(x) be the error in the n'th partial sum.
• Enter in Question 1 the maximum value of |e19(x)| for the triangular wave (correct to 2 significant digits).
• Enter in Question 2 the maximum value of s19(x) for the square wave (correct to 3 significant digits).
• Enter in Question 3 some n for which |en(x)| < 0.01 for all x for the triangular wave. Your answer must be within 20% of the least such n.
• Enter in Question 4 some n such that for the square wave, |em(0.1)| < 0.01 for all m > n. Your answer must be within 20% of the least such n.
• Enter in Question 5 the maximum value of sn(x) for the square wave and the n from Question 4 (correct to 3 significant digits).

## Submission

There is no reason in principle that you will not be able to submit your answers from home, but we cannot guarantee success. If a submission from within the Mathematics Department system is not successful, tell the TA.

## Comment?

Send questions about problems with submitter to the lab TA

Send mathematical questions about the labs to the professor in charge of Math 256: fournier@math.ubc.ca