8.5 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

A Specific Power Series

cf. Stewart, Section 8.5, Example 5, page 604

Let's find the radius of convergence and interval of convergence of a specific power series.

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The Ratio Test is the appropriate instrument for determining the radius of convergence of the power series with nth term a[n].

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The ratio test takes the limit of this ratio as n → ∞, and compares the result to the number 1.

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Since x is real, we require |x|/3 < 1, or |x| < 3. The radius of convergence is 3.

Let's check for convergence at the endpoints of the interval of convergence.

When x = 3, a[n] = n/3, and the series diverges because its nth term does not approach zero.

When x = -3, a[n] = n/3, and once again the series diverges since its nth term does not approach zero.

We conclude that our series converges on the open interval (-3,3) and diverges elsewhere.

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For some reason, Mathematica gets confused when trying to sum this infinite series for negative values of x, so we will plot its graph only for positive x's.

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Restricting the range of f(x) gives a little more information on its behaviour near the x-axis.

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Created by Mathematica (May 5, 2004)