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Prove the ratio test in the following steps. Assume that ) `ak satisfies ap + 0. Assume furthermore k=1 that ak+1 lim =r <1. ak (a) Let q be such that r < q < 1. Explain why there is some N such that n > N implies that |an+1| < lan|· q. (b) Explain why |an|·q* necesarily converges. k=1 (c) Use (b) to prove that >ak| converges, i.e., that > ak converges absolutely. IM:

Prove the ratio test in the following steps. Assume that ) `ak satisfies ap + 0. Assume furthermore k=1 that ak+1 lim =r <1. ak (a) Let q be such that r < q < 1. Explain why there is some N such that n > N implies that |an+1| < lan|· q. (b) Explain why |an|·q* necesarily converges. k=1 (c) Use (b) to prove that >ak| converges, i.e., that > ak converges absolutely. IM:

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section1.2: Exponents And Radicals
Problem 90E
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Prove the ratio test in the following steps. Assume that ) `ak satisfies ak # 0. Assume furthermore k=1 that Ak+1 lim = r <1. ak (a) Let q be such that r < q < 1. Explain why there is some N such that n > N implies that |an+1|< |an| · q. (b) Explain why > JaN|· q* necesarily converges. k=1 (c) Use (b) to prove that >|a| converges, i.e., that >ak converges absolutely.
Transcribed Image Text:Prove the ratio test in the following steps. Assume that ) `ak satisfies ak # 0. Assume furthermore k=1 that Ak+1 lim = r <1. ak (a) Let q be such that r < q < 1. Explain why there is some N such that n > N implies that |an+1|< |an| · q. (b) Explain why > JaN|· q* necesarily converges. k=1 (c) Use (b) to prove that >|a| converges, i.e., that >ak converges absolutely.
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