RedAssign13[1].3 - number that can be included in an...

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Unit: Sequences and Series Module: Power Series Interval and Radius of Convergence [email protected] Copyright 2001, Thinkwell Corp. All Rights Reserved. 1623 –rev 06/15/2001 The interval of convergence of a power series is the collection of points for which the series converges. The radius of convergence of a power series is the distance between the center and either endpoint of the interval of convergence. Use the ratio test to find the radius and interval of convergence. There are four kinds of intervals involving endpoints that are finite only. If an endpoint is not included in the interval, a parenthesis is used. If it is included, a square bracket is used. For intervals with endpoints that are infinite, parentheses are always used. Infinity is not a
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Unformatted text preview: number that can be included in an interval. This is a geometric series. It converges if | x | < 1. The series—and therefore the interval of convergence—is centered about x = 0. You can always use the ratio test to determine the interval of convergence of a power series . For this series, ρ is equal to | x |. If | x | < 1, then the series converges. So the series converges on the interval (–1, 1). Don’t forget to study the endpoints. For x = 1, the sum is infinite. For x = –1, the sum alternates between zero and one. So the series diverges for both endpoints. The interval of convergence is just (–1, 1). The center of this interval is zero. The radius of convergence , which is the distance from the center to either endpoint, is one....
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This note was uploaded on 04/27/2010 for the course MATH 172 taught by Professor Hyon during the Spring '10 term at Community College of Denver.

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