Formal Taylor Series
	with Mathematica

by
K. D.  Stroyan
University of Iowa

copyright 1997 by Academic Press, Inc. - All rights reserved.

Notebook Overview

Mathematica can calculate Taylor Series expansions of functions to any number of terms.  The command to do this is:

Series[ f[x], {x, a, n} ]

where:
	f[x] is the function you wnat to expand in a Taylor series,
	a is the point at which the Taylor series is expanded, 
	n is the number of terms you want to see.

References to the Text

This NoteBook goes with Chapter 26, especially Exercise 26.1.6 of the text.

Computations with Series[.]

A General Expansion at x0 = 0:

Clear[f, x]; Series[f[x], {x, 0, 9}]

The O[x]^10 at the end of the series just tells you that terms of order higher than 10 have been omitted.

The Exponential Series

eSeries[x_] = Series[Exp[x], {x, 0, 9}]

Conversion to "Normal" Expressions

(1/2) (eSeries[I x] + eSeries[-I x])

Normal[(1/2) (eSeries[I x] + eSeries[-I x])]

The Cosine Series

cSeries[x_] = Series[Cos[x], {x, 0, 9}]

Normal Expressions

(1/2) (eSeries[I x] + eSeries[-I x])

Normal[cSeries[x]] == 
      Normal[(1/2) (eSeries[I x] + eSeries[-I x])]

The Sine Series

sSeries[x_] = Series[Sin[x], {x, 0, 9}]

Practice Exercise

Use Normal forms of series to show that
	
	Sin[ x ] = (Exp[ I x ] - Exp[ -I x ])/(2 I)

The Geometric Series

Series[(1/(1 - x)), {x, 0, 9}]

The ArcTan Series

Series[ArcTan[x], {x, 0, 9}]

Expansions at x0 = a

Clear[f, x, a]; Series[f[x], {x, a, 9}]

Two Views of the Log Series at x0 = 1

Series[Log[x], {x, 0, 10}]    
      Series[Log[1 + x], {x, 0, 10}]Series[Log[x], {x, 1, 10}]

Exercises

The Binomial Series (Problem 26.1)

Expand f[x] = (((1 + x))^p) in a series at x0 = 0:

Clear[f, x, p]; f[x_] := (((1 + x))^p)Series[f[x], {x, 0, 9}]

Exercise 26.1.6 The Series for Exp[-x^2]

Find the series expansin for  Exp[ -x^2 ] at  x0 = 0.