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The series is absolutely convergent on the following interval. x-5 <1 |x - 5] < 10 x € (-5, 15) An alternating series is the series with the terms, in which the absolute value of all the terms decreases, is always a convergent series. An infinite series which is always divergent is called the harmonic series.

The series is absolutely convergent on the following interval. x-5 <1 |x - 5] < 10 x € (-5, 15) An alternating series is the series with the terms, in which the absolute value of all the terms decreases, is always a convergent series. An infinite series which is always divergent is called the harmonic series.

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 81E
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The center of the interval is a which equals 5 in this problem. Why would the radius of convergence be 15? for that to be true wouldn't the interval of convergence be (-10, 20) instead? I thought that R would equal 10.

The series is absolutely convergent on the following interval. x-5|<1 |x5| < 10 x € (-5, 15) An alternating series is the series with the terms, in which the absolute value of all the terms decreases, is always a convergent series. An infinite series which is always divergent is called the harmonic series. Atx = -5, the series becomes Σ (-1) (-10) = 1 which is divergent by nth term test. k=1 10k k=1 10k 10k At x = 15, the series becomes Σ(-1)*. n=1 = 00 k=1 (-1)* which is divergent by nth term test. 00 (-1)k Hence, the series > -(x - 5) converges on (-5, 15) and the radius of convergence is 15. 10k whyp
Transcribed Image Text:The series is absolutely convergent on the following interval. x-5|<1 |x5| < 10 x € (-5, 15) An alternating series is the series with the terms, in which the absolute value of all the terms decreases, is always a convergent series. An infinite series which is always divergent is called the harmonic series. Atx = -5, the series becomes Σ (-1) (-10) = 1 which is divergent by nth term test. k=1 10k k=1 10k 10k At x = 15, the series becomes Σ(-1)*. n=1 = 00 k=1 (-1)* which is divergent by nth term test. 00 (-1)k Hence, the series > -(x - 5) converges on (-5, 15) and the radius of convergence is 15. 10k whyp
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