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Theorem 21. The set of x values for which the power series Cn(x – a)" converges can n=0 always be viewed as an interval centered at a. The types of intervals fall into three classes. 1. The interval contains a single point [a, a], and the radius of convergence is 0. 2. The interval is the entire real line. One representation of the real line is (-∞, ), and the radius of convergence is . 3. The interval has a finite positive radius R. There are four possible sub-cases: [a – R, a + R] (a – R, a + R] [a – R, a + R) (a – R, a + R) х2п+1 [+i (2n + 1)! 4. Σ(-1)7-1 n=0

Theorem 21. The set of x values for which the power series Cn(x – a)" converges can n=0 always be viewed as an interval centered at a. The types of intervals fall into three classes. 1. The interval contains a single point [a, a], and the radius of convergence is 0. 2. The interval is the entire real line. One representation of the real line is (-∞, ), and the radius of convergence is . 3. The interval has a finite positive radius R. There are four possible sub-cases: [a – R, a + R] (a – R, a + R] [a – R, a + R) (a – R, a + R) х2п+1 [+i (2n + 1)! 4. Σ(-1)7-1 n=0

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.3: Geometric Sequences
Problem 44E
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Find the interval of convergence and the radius of convergence for each series.

Theorem 21. The set of x values for which the power series Cn(x – a)" converges can n=0 always be viewed as an interval centered at a. The types of intervals fall into three classes. 1. The interval contains a single point [a, a], and the radius of convergence is 0. 2. The interval is the entire real line. One representation of the real line is (-∞, ), and the radius of convergence is . 3. The interval has a finite positive radius R. There are four possible sub-cases: [a – R, a + R] (a – R, a + R] [a – R, a + R) (a – R, a + R)
Transcribed Image Text:Theorem 21. The set of x values for which the power series Cn(x – a)" converges can n=0 always be viewed as an interval centered at a. The types of intervals fall into three classes. 1. The interval contains a single point [a, a], and the radius of convergence is 0. 2. The interval is the entire real line. One representation of the real line is (-∞, ), and the radius of convergence is . 3. The interval has a finite positive radius R. There are four possible sub-cases: [a – R, a + R] (a – R, a + R] [a – R, a + R) (a – R, a + R)
х2п+1 [+i (2n + 1)! 4. Σ(-1)7-1 n=0
Transcribed Image Text:х2п+1 [+i (2n + 1)! 4. Σ(-1)7-1 n=0
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