130 CHAPTER 4. SEQUENCES AND FUNCTIONS OF A DISCRETE VARIABLE

Transcription

1 130 CHAPTER 4. SEQUENCES AND FUNCTIONS OF A DISCRETE VARIABLE

2 Chapter 4 Sequences and functions of a discrete variable 4.1 Introduction In some instances, one may prefer to think of functions where the independent variable is a discrete variable (e.g. a variable which can only be an integer). The dependent variable, on the other hand, can be any real number: Examples: Bald Eagles in the state of Washington are migratory bird that come back every year to nest. It makes sense to count the number of nests at the same date every year to establish the bird population. 131

3 132 CHAPTER 4. SEQUENCES AND FUNCTIONS OF A DISCRETE VARIABLE One can also simply define functions of a discrete variable: let then f(n) = n These functions of a discrete variable form a sequence of numbers. They are often written in the following form: Example: If we consider the sequence of numbers a n = 2 n, then 4.2 Example of sequences: exponential population growth Textbook page Bacteria reproduce by cellular division at regular time intervals (about 20 minutes). Cellular division implies that every bacteria splits into 2 bacteria so that if we start with 1 individual in the petri dish at time t=0, and observe the dish every 20 minutes, we are likely to observe the following number of bacteria : (a) t=20 (b) t=40 min (c) t=60 min

4 4.3. IMPLICIT RULES 133 Observation round number: Time: Number of bacteria in the dish: So, if we write the number of bacteria in the dish at the n-th observation as a n, then we observe that the sequence of numbers a n follow two (equivalent) rules: Both rules describe the same sequence of numbers; one of the rules describes it explicitly: the other one describes it implicitly: From the explicit expression of the sequence, we see that bacterial growth is exponential. See Handout 4.3 Implicit rules Definition Examples of implicit rules (i) IQ tests often involve implicit rules: see handout

5 134 CHAPTER 4. SEQUENCES AND FUNCTIONS OF A DISCRETE VARIABLE 3, 7, 15, 31,...: 3, 15, 63,...: (ii) Banking: Suppose a Bank offers a savings account with a 5% interest rate but charges $20 per year. If you open the account with a 0 dollars, what is the amount of money in your account a n after n years? What is the minimum amount of money you should open the bank account with?

6 4.4. LIMITS OF SEQUENCES Cobweb diagrams The cobweb diagram is a tool for vizualising the progression of an implicit sequence as n increases. To construct a cobweb diagram: Draw the axes: a n is the x axis, a n+1 is the y axis Draw the a n+1 = a n line (i.e. the x = y line) with a dashed-line. Draw very accurately the graph of the function a n+1 = f(a n ) as a solid line Place a 0 on the x-axis Trace a vertical line from a 0 to the solid line Trace a horizontal line from there to the dashed line; the x coordinate of this point is the next in the sequence, a 1. Go vertically to the solid line Go horizontally to the dashed line; the x coordinate of this point is the next in the sequence, a 2. etc... Examples: See Handouts a n+1 = a 2 n a n+1 = 2.5a n (1 a n ) Note: The graph of the function a n+1 = f(a n ) must be very accurate, otherwise the cobweb diagram will be misleading. The behavior of the sequence depends crucially on the value of a 0. The cobweb method provides a visual way of finding out the behavior of the sequence as n tends to infinity. 4.4 Limits of sequences Textbook pages In many situations, we are interested in the behavior of a sequence as n. For instance, if the sequence a n represents the number of individuals in a population (or the value of your portofolio), we are interested in knowing whether

7 136 CHAPTER 4. SEQUENCES AND FUNCTIONS OF A DISCRETE VARIABLE This section extends the concept of limits to sequences, and how to find them for explicit and implicit sequences Limits of explicit sequences The limit as n of an explicitly defined sequence is entirely analogous to the limit as x + of functions. In fact: Examples: What are the limits of the following sequences? lim n a n = 3n2 2 n 2 +2n 1 lim n a n = 2 n+1 1 lim n a n = ln(n) n Limits of recursively defined sequences When a sequence is defined recursively, a n+1 = f(a n ) it is often more difficult to guess what the limit of the sequence will be as n. However, the cobweb diagrams suggest the answer to the problem. Sequences converging to a finite value: Whenever the sequence appeared to converge to a limit, that limit points lies at the intersection of the a n+1 = a n line (dashed line), and the a n+1 = f(a n ) line(solid line).

8 4.4. LIMITS OF SEQUENCES 137 Examples: an+1 a n+1 = f(a n ) a n+1 =a n an+1 a n+1 =a n a n+1 = f(a n ) a3 a2 a1 a0 a n a0 a2 a4 a5 a3 a1 a n (d) (e) Definition: Mathematical equation for the fixed points of an implicit sequence: Examples: (see Handout on Cobweb diagrams) a n+1 = a 2 n

9 138 CHAPTER 4. SEQUENCES AND FUNCTIONS OF A DISCRETE VARIABLE a n+1 = 2.5a n (1 a n ) Note: Although the limit of a sequence is necessarily one of the fixed points of the recursion, the converse is not true: sometimes, the fixed point is not a limit. Examples: Sequences diverging to + or : In some cases, the sequence does not converge to the fixed points but instead diverges to either + or. This can be readily seen in the cobweb diagram as an ever-ascending or ever-descending staircase :

10 4.4. LIMITS OF SEQUENCES 139 Sequences neither diverging nor converging: Finally, in some cases we find that sequences can alternate between two, or a finite set of points in a periodic manner, or even appear to wander around erratically forever. The latter is an example of chaos. an+1 a n+1 = f(a n ) a n+1 =a n a n+1 a n+1 =a n an+1 = f(a n) an a0 a2 (f) a1 a3 a n (g)

11 140 CHAPTER 4. SEQUENCES AND FUNCTIONS OF A DISCRETE VARIABLE Check your understanding of Lecture 18 Explicit sequences: For each of the following sequences of numbers, work out the explicit formula a n = f(n) they satisfy: 1 2, 2 3, 3 4, 4 5,... 1, (1 1/2) 1, (1 2/3) 1, (1 3/4) 1, (1 4/5) 1,... sin 1, 2 sin(1/2), 3 sin(1/3), 4 sin(1/4),... Implicit sequences: For each of the following implicit sequences: a 0 = 1, a n+1 = 3a n a 0 = 1, a n+1 = 3a n 2 a 0 = 1, a n+1 = a n + 1 a n (1) Write the first 5 terms of the sequences; (2) plot the sequence graphically on a cobweb diagram as was shown in the lectures; (3) what do the numbers a n tend to when n tends to infinity? Limits of explicit sequences: Textbook page 114 Problems 1, 2, 4, 7. Find all the fixed points of these recursively defined sequences. Using a cobweb diagram, find towards which of the fixed points the sequence is converging. a n+1 = 0.5(a n + 5), a 0 = 1 a n+1 = 2a n, a 0 = 1 a n+1 = 2a n (a n 1), a 0 = 0.1