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/