Chaos Theory and Converging SequencesDate: 2/14/96 at 10:32:10 From: sarah safraz Subject: sequences Hi. I'm emailing from London. I'm a 28-year-old having to resit a math exam! As part of my course work I'm investigating a process of dividing and adding (i.e. take a number, divide by 5, add 2, write down result, repeat process using the result). I realise that using this particular process my sequence will converge towards 2.5, where nothing else can be done to it. However, by reversing the process, i.e. adding 2 first then dividing by 5, my sequence will diverge to what I guess will be infinity. Trouble is, I don't know what else to say about this investigation!! Can this really be all there is to do? Could I do anything to jazz it up a bit? Yours in frustration, Billy (Sarah) Burlingham Date: 2/14/96 at 11:17:4 From: Doctor Byron Subject: Re: sequences Wow! What a great observation! You may not realize it, but the study of sequences like the one you just mentioned has become one of the hottest topics in mathematics in recent years. Mathematicians have found sequences like this that diverge, converge, jump back and forth between different numbers, and even oscillate in wild chaotic patterns. Speaking of chaos, you may have heard the term 'chaos theory' before (perhaps the movie Jurassic Park?). One of the more famous examples of chaos theory is a sequence like the one you just described. Choose some constant, C, and an initial value x1. Now create a new number x2 according to the equation: x2 = C(x1)(1-x1) [This is called the "logistic equation"] Depending on your choice of C this equation can have wildly different effects.In most cases, the behavior of the system is not dependent on the initial choice of x1 (as you noted in your case with x/5 + 2). For certain values of C, however, the behavior of the system in wild and chaotic (leading to some great computer graphics, I might add). Actually, the chaotic behavior starts somewhere around C > 3.57. For C = 3.0 you get a system that will eventually oscillate between four states. If you are interested in learning more about Chaos Theory, there is a book called "Chaos" by James Gleik which is very readable and manages to discuss the subject without too much heavy mathematics. To get back to your question of why x2 = (x1)/5 + 2 converges to 2.5, however, this sequence can be rewritten in a generalexpression of the form: xn = [(.2)^n]x1 + [2 + (.2)2 + (.2)^2*2 + (.2)^3*2 ... + (.2)^n*2] The first expression, which depends on x1, gets smaller and smaller as n gets larger until it is essentially insignificant. You will notice,however that the second long expression in terms of 2 and powers of 0.2 does not depend on x1. It is possible to show that this series has an sum which gets closer and closer to 2.5 as n approaches infinity. This is why you observed the behavior you did. Good luck with your math exam. This is a very rich topic, and I hopeyou enjoy exploring it! -Doctor Byron, The Math Forum |
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