8.6 Representation of Functions as Power Series

Mathematica script by Chris Parrish,
   cparrish@sewanee.edu

Sources and references for some of these problems include
    James Stewart, "Calculus: Concepts and Contexts," Second Edition, Brooks/Cole, 2001
    Deborah Hughes-Hallett, Andrew M. Gleason, et. al., "Calculus," Second Edition, John Wiley & Sons, 1998
    Robert Fraga, ed., "Calculus Problems for a New Century," The Mathematical Association of America, 1993
    Selwyn Hollis, "CalcLabs with
Mathematica" for Stewart's "Single Variable Calculus, Concepts and Contexts, Second Edition," Brooks/Cole, 2001

Representing a Specific Function as a Power Series

cf. Stewart, Section 8.6, Example 8, page 609

Let's start with a specific function.

In[9]:=

<<Graphics`Colors`

Clear[f, x, fSeries] ;  f[x_] := 1/(1 + x^7)                                   ...                                   7                                                      1\ + \ x

[Graphics:HTMLFiles/8.6_functionRepresentation_3.gif]

What is the integral of this function?

In[13]:=

∫f[x] x

Out[13]=

1/7 (Log[1 + x] - Cos[π/7] Log[1 + x^2 - 2 x Cos[π/7]] - Cos[(3 π)/7] Log[1 + x ... 0;)/7]] Sin[(3 π)/7] + 2 ArcTan[(x - Cos[(5 π)/7]) Csc[(5 π)/7]] Sin[(5 π)/7])

Wow! We could perhaps work with that, but it will turn out to be much simpler to express this function as a power series, and then work with that power series representation.

In[14]:=

Series[f[x], {x, 0, 30}] fSeries[x_] = Normal[%]

Out[14]=

1 - x^7 + x^14 - x^21 + x^28 + O[x]^31

Out[15]=

1 - x^7 + x^14 - x^21 + x^28

The function f has a singularity at x = -1, but its power series representation is a polynomial, hence has no singularities whatsoever. So just how similar are these two functions?

In[16]:=

Plot[{f[x], fSeries[x]}, {x, -3, 3}, PlotRange {-1, 2}, PlotLabel-> ... xesLabel {"x", None}, PlotStyle {LightCadmiumRed, TerreVerte}] ;

[Graphics:HTMLFiles/8.6_functionRepresentation_10.gif]

The power series representation of f seems to be doing a very good job of tracking f over the interval (-1,1), but it is hopeless outside of this interval. The theory developed in Stewart, Section 8.6, explains this behaviour.


Created by Mathematica  (May 5, 2004)