Problem 59. Let b be a nonzero real number with |b| < 1 and let E > 0. (a) Solve the inequality |b|" < e for n (b) Use part (a) to prove lim,b" = 0.

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
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#59 part b
9:11
( Safari
RealAnalysis-ISBN-fix...
n+4
Hence by definition, lim
= 0.
n00 n2 +1
DE
Again we emphasize that the scrapwork is NOT part of the formal proo:
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 e > 0.
(a) Solve the inequality |b|" < e for n
(b) Use part (a) to prove lim,n→∞b" = 0.
CONVERGENCE OF SEQUENCES AND SERIES
7
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)")-o = (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
Conronging to none mesrna thet anu time a
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Transcribed Image Text:9:11 ( Safari RealAnalysis-ISBN-fix... n+4 Hence by definition, lim = 0. n00 n2 +1 DE Again we emphasize that the scrapwork is NOT part of the formal proo: 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 e > 0. (a) Solve the inequality |b|" < e for n (b) Use part (a) to prove lim,n→∞b" = 0. CONVERGENCE OF SEQUENCES AND SERIES 7 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)")-o = (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 Conronging to none mesrna thet anu time a Next > Dashboard Calendar To Do Notifications Inbox 因
Expert Solution
Step 1

We have to show that limn bn=0 where b is a nonzero real number with

b<1.

We know that limn xn=L, where L is a real number, if and only if for ε>0

there exists a positive integer m such that xn-L<ε for all nm.

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