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00 Proof of the Root Test Let S => an be a positive series, and assume that L = lim Van 63. exists. (a) Show that S converges if L < 1. Hint: Choose R with L < R < 1 and show that a, < R" for n sufficiently large. Then compare with the geometric series R". (b) Show that S diverges if L > 1. SOLUTION Suppose lim a, = L exists. n=0

00 Proof of the Root Test Let S => an be a positive series, and assume that L = lim Van 63. exists. (a) Show that S converges if L < 1. Hint: Choose R with L < R < 1 and show that a, < R" for n sufficiently large. Then compare with the geometric series R". (b) Show that S diverges if L > 1. SOLUTION Suppose lim a, = L exists. n=0

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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00 Proof of the Root Test Let S => an be a positive series, and assume that L = lim Van 63. exists. (a) Show that S converges if L < 1. Hint: Choose R with L < R < 1 and show that a, < R" for n sufficiently large. Then compare with the geometric series R". (b) Show that S diverges if L > 1. SOLUTION Suppose lim a, = L exists. n=0
Transcribed Image Text:00 Proof of the Root Test Let S => an be a positive series, and assume that L = lim Van 63. exists. (a) Show that S converges if L < 1. Hint: Choose R with L < R < 1 and show that a, < R" for n sufficiently large. Then compare with the geometric series R". (b) Show that S diverges if L > 1. SOLUTION Suppose lim a, = L exists. n=0
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