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.

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

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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?

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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.

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

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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)