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b) an unbounded sequence with a convergent subsequence.

b) an unbounded sequence with a convergent subsequence.

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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**b) An Unbounded Sequence with a Convergent Subsequence** In mathematics, a sequence can be described as a list of numbers in a specific order. A sequence is said to be "unbounded" if its terms do not have a finite upper or lower limit. Despite being unbounded, such sequences can still contain "convergent subsequences." A convergent subsequence is a subset of the overall sequence whose terms approach a specific value as the sequence progresses. For instance, consider the sequence of terms like \( a_n = (-1)^n n \) where the sequence alternates and grows without bound: \(-1, 2, -3, 4, -5, \ldots\). This sequence is unbounded because it increases indefinitely in magnitude. However, by selecting specific terms, such as all even-indexed terms \(2, 4, 6, \ldots\), a convergent subsequence can be identified, wherein these terms approach infinity in a more controlled manner.
Transcribed Image Text:**b) An Unbounded Sequence with a Convergent Subsequence** In mathematics, a sequence can be described as a list of numbers in a specific order. A sequence is said to be "unbounded" if its terms do not have a finite upper or lower limit. Despite being unbounded, such sequences can still contain "convergent subsequences." A convergent subsequence is a subset of the overall sequence whose terms approach a specific value as the sequence progresses. For instance, consider the sequence of terms like \( a_n = (-1)^n n \) where the sequence alternates and grows without bound: \(-1, 2, -3, 4, -5, \ldots\). This sequence is unbounded because it increases indefinitely in magnitude. However, by selecting specific terms, such as all even-indexed terms \(2, 4, 6, \ldots\), a convergent subsequence can be identified, wherein these terms approach infinity in a more controlled manner.
2. Give examples (without proofs) of the following: a) sequences \(\{x_n\}\) and \(\{y_n\}\) which both diverge, but whose sum \(\{x_n + y_n\}\) converges.
Transcribed Image Text:2. Give examples (without proofs) of the following: a) sequences \(\{x_n\}\) and \(\{y_n\}\) which both diverge, but whose sum \(\{x_n + y_n\}\) converges.
Expert Solution
Step 1

A sequence (x n) of reals is bounded iff there is some M≥0 such that |x n|≤M for all n≥1.

A sequence which is not bounded is called unbounded sequence.

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