**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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Answered: Show directly from the definition that…

Math

Advanced Math

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

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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Question

**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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