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Problem 57. Use the definition of convergence to zero to prove the following. (a) lim = 0

Problem 57. Use the definition of convergence to zero to prove the following. (a) lim = 0

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.1: Infinite Sequences And Summation Notation
Problem 72E
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8:58 ( Safari RealAnalysis-ISBN-fix... CONVERGENCE OF SEQUENCES AND SERIES 76 Problem 57. Use the definition of convergence to zero to prove the following. 1 (a) lim = 0 n o n2 1 (b) lim = 0 n0 Vn As the sequences get more complicated, doing scrapwork ahead of time will become more necessary. Example 4. Use the definition of convergence to zero to prove n+4 = 0. lim no n2 + 1 SCRAPWORK: Given an e > 0, we need to see how large to make n in order to guarantee that | < e. First notice that < 4. Also, notice that if n > 4, then n +4 < n+n = 2n. So as long as n > 4, we have < < 2 = 2. We can make this less than ɛ if we make n > ?. This means we need to make n > 4 and n > ?, simultaneously. These can be done if we let N be the maximum of these two numbers. This sort of thing comes up regularly, so the notation N = max (4, 2) was developed to mean the marimum of these two numbers. Notice that if N = max (4, 2) then N > 4 and N > ?. We're now ready for the formal proof. END OF SCRAPWORK Proof: Let e > 0. Let N = max (4, 2). If n > N, then n > 4 and n> ?. Thus we have n >4 and 2 < e. Therefore n+4 n+4 n +4 2n n2 +1 n² + 1 n2 n2 < E. n n +4 Hence by definition, lim = 0 no n2 +1 Again we emphasize that the scrapwork is NOT part of the formal proof and the reader will not see it. However, if you look carefully, you can see the scrapwork in the formal proof. Problem 58. Use the definition of convergence to zero to prove n2 + 4n +1 lim = 0. n3 Problem 59. Let b be a nonzero real number with |b| <1 and let ɛ > 0. (a) Solve the inequality |b|" < e for n Next Dashboard Calendar To Do Notifications Inbox
Transcribed Image Text:8:58 ( Safari RealAnalysis-ISBN-fix... CONVERGENCE OF SEQUENCES AND SERIES 76 Problem 57. Use the definition of convergence to zero to prove the following. 1 (a) lim = 0 n o n2 1 (b) lim = 0 n0 Vn As the sequences get more complicated, doing scrapwork ahead of time will become more necessary. Example 4. Use the definition of convergence to zero to prove n+4 = 0. lim no n2 + 1 SCRAPWORK: Given an e > 0, we need to see how large to make n in order to guarantee that | < e. First notice that < 4. Also, notice that if n > 4, then n +4 < n+n = 2n. So as long as n > 4, we have < < 2 = 2. We can make this less than ɛ if we make n > ?. This means we need to make n > 4 and n > ?, simultaneously. These can be done if we let N be the maximum of these two numbers. This sort of thing comes up regularly, so the notation N = max (4, 2) was developed to mean the marimum of these two numbers. Notice that if N = max (4, 2) then N > 4 and N > ?. We're now ready for the formal proof. END OF SCRAPWORK Proof: Let e > 0. Let N = max (4, 2). If n > N, then n > 4 and n> ?. Thus we have n >4 and 2 < e. Therefore n+4 n+4 n +4 2n n2 +1 n² + 1 n2 n2 < E. n n +4 Hence by definition, lim = 0 no n2 +1 Again we emphasize that the scrapwork is NOT part of the formal proof and the reader will not see it. However, if you look carefully, you can see the scrapwork in the formal proof. Problem 58. Use the definition of convergence to zero to prove n2 + 4n +1 lim = 0. n3 Problem 59. Let b be a nonzero real number with |b| <1 and let ɛ > 0. (a) Solve the inequality |b|" < e for n Next Dashboard Calendar To Do Notifications Inbox
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