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Show directly from the definition that the following are Cauchy sequences: ^(N+1) B (1 + 2/₁ + ... + NT)

Show directly from the definition that the following are Cauchy sequences: ^(N+1) B (1 + 2/₁ + ... + NT)

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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**Title: Proving Cauchy Sequences** **Objective:** Show directly from the definition that the following are Cauchy sequences: **A:** \(\left( \frac{N+1}{N} \right)\) **B:** \(\left( 1 + \frac{1}{2!} + \cdots + \frac{1}{N!} \right)\) **Explanation:** 1. **Sequence A**: This expression \(\left( \frac{N+1}{N} \right)\) suggests that as \(N\) approaches infinity, the sequence is described by the ratio of \(N+1\) over \(N\). 2. **Sequence B**: This sequence is expressed as a series, starting from \(1\) and including fractions with factorials as denominators up to \(\frac{1}{N!}\). **Definition Reminder:** A *Cauchy sequence* is a sequence where, for every positive number \(\epsilon\), there exists an integer \(M\) such that for all integers \(m, n > M\), the absolute difference \(|a_n - a_m| < \epsilon\). For each sequence, demonstrate this property by evaluating the limits and verifying they fulfill the conditions specified in the definition.
Transcribed Image Text:**Title: Proving Cauchy Sequences** **Objective:** Show directly from the definition that the following are Cauchy sequences: **A:** \(\left( \frac{N+1}{N} \right)\) **B:** \(\left( 1 + \frac{1}{2!} + \cdots + \frac{1}{N!} \right)\) **Explanation:** 1. **Sequence A**: This expression \(\left( \frac{N+1}{N} \right)\) suggests that as \(N\) approaches infinity, the sequence is described by the ratio of \(N+1\) over \(N\). 2. **Sequence B**: This sequence is expressed as a series, starting from \(1\) and including fractions with factorials as denominators up to \(\frac{1}{N!}\). **Definition Reminder:** A *Cauchy sequence* is a sequence where, for every positive number \(\epsilon\), there exists an integer \(M\) such that for all integers \(m, n > M\), the absolute difference \(|a_n - a_m| < \epsilon\). For each sequence, demonstrate this property by evaluating the limits and verifying they fulfill the conditions specified in the definition.
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How do you keep flipping inequalities to end up at =2/m?

Also, if it Xn - Xm, wouldnt it be (1+1/n) - (1+1/m)??

Where does the 1+1/n - (1-1/m) come from?

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