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]:=

What is the integral of this function?

In[13]:=

Out[13]=

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]:=

Out[14]=

Out[15]=

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]:=

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)