For an initial value problem, you would be given the initial terms in the sequence. For a second order problem, you would need two initial values, a₀ and a₁. At this stage you should be thinking this looks a lot like differential equations, and you are right. Many of the ideas we've developed in this class work for both differential and difference equations. Furthermore, just as we have found it useful to analyze solutions to differential equations using series techniques to reduce the situations to questions about Taylor coefficients, it is often useful to analyze a sequence by treating the terms as the coefficients of a Taylor series. In mathematics this technique is called computing the generating function, while in engineering it is referred to as a z-transform (and in statistics as the moment generating function). In this lab, we will explore a little bit about difference equations and their associated generating functions.

1. Launch the difference equation applet. In the applet, you can define a second order constant coefficient difference equation. Initial values are set by filling in text boxes and pressing a button to signal you want the values updated, just as in previous labs. You adjust the coefficients by moving the slidersm which is a different interface than we've used before. The advantage of this interface is that you can slide through many values quickly to get a sense of what the different possibilities are. The downside is that you don't have as fine a control over the values, and you can scroll through the possibilities too quickly so you can find different possibilities without having to think too much (and as your teacher, my interest is in your thinking). But since this isn't a course in difference equations and I just want you to get a basic sense of some of the connections that may come up in later courses, this interface is appropriate here. Below the first 25 terms of the sequence are illustrated in the left graph and the generating function (actually, the degree 100 Taylor polynomial approximation to the generating function) is illustrated in the right graph, with the radius of convergence marked by the red dotted lines.

2. Slide the sliders back and forth and see how the graphs change. You should see situations where the coefficients (as graphed on the left) grow to inifinity, shrink to 0, reach a constant value and stay put, oscillate at constant amplitude, oscillate with increasing amplitude, and oscillate with decreasing amplitude. Write a paragraph that describes how the character of the sequence changes as you vary the P, Q, and R values in turn. By the way, note that the different types of sequences match the different types of solutions we had to constant coefficient second order equations. This is not an accident, though the connections are a bit complicated so I don't expect you to work out all the rules for exactly which values of P, Q, and R correspond to which types of sequences, though any rules you can figure out will improve your paragraph.

3. Next consider how the graph of the generating function is related to graph of the sequence of coefficients. In particular, consider how the radius of convergence changes. As in last week's lab, you can usually identify the radius of convergence easily from the graph, and as in last week's lab any parts of the graph past the radius of convergence should be ignored as garbage. Write a short paragraph describing how the radius of convergence of the generating function is related to the character (increasing, decreasing, etc.) of the sequence of coefficients.

4. You should have noted in the last part that when the sequence of coefficients is increasing (or the amplitude is increasing when the sequence is oscillating), that the radius of convergence is less than 1, while when the radius of convergence is greater than 1 the sequence of coefficients is decreasing (or at least the amplitude of the oscillations is decreasing). Write a paragraph to explain why this is reasonable. You may find it easiest to explain by using the raio test to define the radius of convergence. Or perhaps you will find it easier to explain if you consider the connection between S_n=0^inf a_n and f(1), where f is the generating function.

5. So far we haven't played around much with a₀ and a₁. Try out some different values and see what happens. Based on your experience with second order differential equations, you shouldn't be too surprised when the initial values don't have much effect on the overall character of the sequence of terms. On the other hand, you should see some changes. Write a paragraph that explains how and why changing the values of a₀ and a₁ change the graph of the generating function about the point x = 0.