8.8 Binomial 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

A Specific Binomial Series

cf. Stewart, Section 8.8, Example 1, page 624

Let's start with a specific function whose Taylor series is a binomial series.

In[427]:=

In[428]:=

The power series representation of this function is a particularly simple binomial series.

In[431]:=

Out[431]=

Out[432]=

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

In[433]:=

This particular partial sum of the power series representation of f (a binomial series) seems to be doing a very good job of tracking f over much of the interval (-1,1), but it is hopeless outside of this interval. The theory developed in Stewart, Section 8.8, explains this behaviour. The power series representation is a binomial series expansion about x = 0, and it has radius of convergence R = 1.

Another Binomial Series

cf. Stewart, Section 8.8, Example 2, page 624

Let's start with another specific function whose Taylor series is a binomial series.

In[434]:=

The power series representation of this function is a binomial series.

In[437]:=

Out[437]=

Out[438]=

The function f has a singularity at x = 4, but its approximating polynomial has no singularities whatsoever. So just how similar are these two functions?

In[439]:=

This particular partial sum of the power series representation of f (a binomial series) seems to be doing a very good job of tracking f over much of the interval (-4,4), but it is hopeless outside of this interval. In fact, f is not even defined for x > 4, but the Taylor Polynomial is defined for all x. The theory developed in Stewart, Section 8.8, explains this behaviour. The power series representation is a binomial series expansion about x = 0, and it has radius of convergence R = 4.

Created by Mathematica  (May 5, 2004)