Why Study Sequences and Series?Date: 02/26/99 at 12:52:15 From: Tara Leiviska Subject: Purpose of Arithmetic Sequences and Series I am teaching arithmetic sequences and series to 10th graders. They keep asking me why they need to know sequences and series. I have no idea why. I have never been told myself. I cannot find a Web site or a book that will tell me some applications of arithmetic sequences or series. Please help! Date: 03/02/99 at 18:52:42 From: Doctor Nick Subject: Re: Purpose of Arithmetic Sequences and Series Sequences and series are useful in the same way geometry is useful. There are things in the world that can be represented by circles and squares, and things that can be represented as sequences and series. For the variety of things that sequences can represent, take a look at Sloane's On-Line Encyclopedia of Integer Sequences: http://www.research.att.com/~njas/sequences/ This is a gigantic collection of sequences that come from all kinds of applications. Take a look at the Fibonacci sequence (a classic) at Ron Knott's site: http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html Series involve one of the most powerful ideas in mathematics. In particular, power series are amazingly useful for doing all kinds of things. One thing they are very useful for is approximations of functions, which is necessary in virtually all computer applications involving evaluations of functions, an interesting computer application. If you are not very familiar with power series, I suggest reading up on them. Another kind of series that is amazingly useful is Fourier series. This is a somewhat advanced topic, but certainly could be introduced to willing high school students, at least as an example of the cool things that you can do with series. Fourier series, and ideas connected with them, are used in many many signal processing applications (sound, video, etc.). Ask your students to explain how the bass and treble knobs on their stereos work. Those who do a good job answering the question will run into Fourier series. Another thing to consider is musical synthesis: how does a synthesizer work? The answer involves Fourier series (along with other interesting materials, of course). For some neat stuff related to Fourier series (Fourier transforms, guitars, and drums), take a look around Dan Russell's page on Vibration and Waves Animations: http://www.gmi.edu/~drussell/Demos.html Series are useful as a way of creating functions. Once you go through rational, exponential, and trigonometric functions, what is next? Functions created from series, with power series and Fourier series being two of the more popular kinds. One place such functions arise is as the solutions to differential equations. In probability, one runs into series quite a bit. Consider the following problem. You throw a 6-sided die until either a six comes up, and you win, or a one comes up, and you lose. By symmetry, it is clear that you have a 50% chance of winning. Another way to look at it is the following. The probability that you will win in exactly n throws of the die is ((4/6)^(n-1)) * (1/6) So, the probability that you will win is the infinite series, where we sum the above expression as n runs from 1 to infinity. This is a nice geometric series, and its sum is (as it has to be) 1/2. Now, in a probability problem, where we do not have this nice kind of symmetry, the series may be the only way to go. (Even in these simpler cases, it is always nice to have more than one way to solve a problem). I could go on and on. Series are used in many, many applications. Have fun, - Doctor Nick, The Math Forum http://mathforum.org/dr.math/ |
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