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### What are Taylor series?

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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

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
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/
```
Associated Topics:
High School Sequences, Series

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