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Problem 59. Let b be a nonzero real number with |b| <1 and let e > 0. (a) Solve the inequality |b|" < e for n

Problem 59. Let b be a nonzero real number with |b| <1 and let e > 0. (a) Solve the inequality |b|" < e for n

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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9:14 1 Safari RealAnalysis-ISBN-fix.. Problem 59. Let b be a nonzero real number with |b| < 1 and let ɛ > 0. (a) Solve the inequality |b|" < e for n (b) Use part (a) to prove lim,t b" = 0. CONVERGENCE OF SEQUENCES AND SERIES 77 We can negate this definition to prove that a particular sequence does not converge to zero. Example 5. Use the definition to prove that the sequence (1+(-1)")0 = (2,0, 2, 0, 2, .) does not converge to zero. Before we provide this proof, let's analyze what it means for a sequence (sn) to not converge to zero. Converging to zero means that any time a distance e >0 is given, we must be able to respond with a number N such that |s,| < E for every n > N. To have this not happen, we must be able to find some e >0 such that no choice of N will work. Of course, if we find such an ɛ, then any smaller one will fail to have such an N, but we only need one to mess us up. If you stare at the example long enough, you see that any e with 0 < e< 2 will cause problems. For our purposes, we will let e = 2. Proof: Let e = 2 and let N EN be any integer. If we let k be any non- negative integer with k > , thenn= 2k > N, but |1 + (–1)"| = 2. Thus no choice of N will satisfy the conditions of the definition for this ɛ, (namely that |1+ (-1)"| < 2 for all n > N) and so lim,0 (1+ (-1)") #0. Problem 60. Negate the definition of lim, $n = 0 to provide a formal definition for lim,+ Sn # 0. Problem 61. Use the definition to prove lim 0 100 #0. Now that we have a handle on how to rigorously prove that a sequence converges to zero, let's generalize this to a formal definition for a sequence Next Dashboard Calendar To Do Notifications Inbox
Transcribed Image Text:9:14 1 Safari RealAnalysis-ISBN-fix.. Problem 59. Let b be a nonzero real number with |b| < 1 and let ɛ > 0. (a) Solve the inequality |b|" < e for n (b) Use part (a) to prove lim,t b" = 0. CONVERGENCE OF SEQUENCES AND SERIES 77 We can negate this definition to prove that a particular sequence does not converge to zero. Example 5. Use the definition to prove that the sequence (1+(-1)")0 = (2,0, 2, 0, 2, .) does not converge to zero. Before we provide this proof, let's analyze what it means for a sequence (sn) to not converge to zero. Converging to zero means that any time a distance e >0 is given, we must be able to respond with a number N such that |s,| < E for every n > N. To have this not happen, we must be able to find some e >0 such that no choice of N will work. Of course, if we find such an ɛ, then any smaller one will fail to have such an N, but we only need one to mess us up. If you stare at the example long enough, you see that any e with 0 < e< 2 will cause problems. For our purposes, we will let e = 2. Proof: Let e = 2 and let N EN be any integer. If we let k be any non- negative integer with k > , thenn= 2k > N, but |1 + (–1)"| = 2. Thus no choice of N will satisfy the conditions of the definition for this ɛ, (namely that |1+ (-1)"| < 2 for all n > N) and so lim,0 (1+ (-1)") #0. Problem 60. Negate the definition of lim, $n = 0 to provide a formal definition for lim,+ Sn # 0. Problem 61. Use the definition to prove lim 0 100 #0. Now that we have a handle on how to rigorously prove that a sequence converges to zero, let's generalize this to a formal definition for a sequence Next Dashboard Calendar To Do Notifications Inbox
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