What are Taylor series?

Date: 04/18/99 at 08:20:05
From: Sandra Springer 
Subject: Taylor Series

What are Taylor Series? What are they needed for? I'm very interested 
in more information about this topic (books, articles, Internet 
addresses,...).

Thank you for your help.

Regards

Sandra Springer

Date: 04/22/99 at 13:13:30
From: Doctor Nick
Subject: Re: Taylor Series

Hi Sandra -

Taylor series are a kind of power series that can be used to give 
representations of functions. A power series implicitly defines a 
functions where it converges; what we often want to do is get a power 
series that represents a function we have in mind. The use of Taylor 
series is one way to generate such a power series. The basic idea is 
that certain functions can be expressed as a series involving their 
derivatives.

One reason Taylor series are important is that they can be used to 
calculate values of a function efficiently. For instance, note that 
the exponential function, e^x, is very nearly equal to 
1+x+(x^2)/2+(x^3)/6+(x^4)/24 for x near zero (say, x between -1 and 
1). This fourth-degree polynomial is the first 5 terms of the Taylor
series for e^x. Taylor series are used in calculators and computers
to calculate things like e^x, cos x, sin x, and so on, for arbitrary
values of x (e.g. e^(0.43), cos(0.311), sin(-3.423)).

You can find lots about Taylor series in just about any book on 
calculus. Most college-level textbooks cover the subject. That's the 
best way to find out more about them. Try your local library, or ask 
math teachers in your school if they can help you find a calculus book 
that covers this subject. Write back if you need more help, or have 
further questions.

Have fun,

- Doctor Nick, The Math Forum
  http://mathforum.org/dr.math/