Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Question 6 (a)
**Determine if the following series converges absolutely, converges conditionally, or diverges. Show your work and justify your answer:**
\[
\sum_{n=1}^{\infty} \left(\frac{5n-1}{7n+2}\right)^n
\]
### Explanation:
This is a typical problem in calculus and real analysis dealing with the convergence of infinite series. The series given is
\[
\sum_{n=1}^{\infty} \left(\frac{5n-1}{7n+2}\right)^n
\]
To determine whether this series converges absolutely, conditionally, or diverges, one would generally follow a structured process involving tests for convergence. For educational purposes, it is important to demonstrate methods such as the Ratio Test, Root Test, or comparison to a known convergent or divergent series.
---
### Steps to Analyze the Series:
1. **Examine the Ratio Test or Root Test**:
These tests are often applied to series where \(a_n\) is of the form \(\left(\frac{a_n}{b_n}\right)^n\). For instance, the Root Test could be particularly useful here because it deals with \(a_n^n\).
2. **Simplification of Terms**:
Simplify the expression \(\left(\frac{5n-1}{7n+2}\right)^n\) for large \(n\):
\[
\frac{5n-1}{7n+2} \approx \frac{5n}{7n} = \frac{5}{7}
\]
3. **Apply the Root Test**:
The Root Test involves taking the \(n\)-th root of \(a_n^n\):
\[
\sqrt[n]{\left(\frac{5n-1}{7n+2}\right)^n} = \left(\frac{5n-1}{7n+2}\right)
\]
4. **Limit Analysis**:
Determine the limit as \(n\) approaches infinity:
\[
\lim_{n \to \infty} \left(\frac{5n-1}{7n+2}\right) = \frac{5}{7}
\]
If this limit is less than 1, the Root Test would indicate that the](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7dbb4ae4-0d65-4baa-9481-63f79be91eca%2F610422cb-ba1b-424f-8be3-a0f78e9d8439%2Faqwz35h_reoriented.jpeg&w=3840&q=75)
Transcribed Image Text:### Question 6 (a)
**Determine if the following series converges absolutely, converges conditionally, or diverges. Show your work and justify your answer:**
\[
\sum_{n=1}^{\infty} \left(\frac{5n-1}{7n+2}\right)^n
\]
### Explanation:
This is a typical problem in calculus and real analysis dealing with the convergence of infinite series. The series given is
\[
\sum_{n=1}^{\infty} \left(\frac{5n-1}{7n+2}\right)^n
\]
To determine whether this series converges absolutely, conditionally, or diverges, one would generally follow a structured process involving tests for convergence. For educational purposes, it is important to demonstrate methods such as the Ratio Test, Root Test, or comparison to a known convergent or divergent series.
---
### Steps to Analyze the Series:
1. **Examine the Ratio Test or Root Test**:
These tests are often applied to series where \(a_n\) is of the form \(\left(\frac{a_n}{b_n}\right)^n\). For instance, the Root Test could be particularly useful here because it deals with \(a_n^n\).
2. **Simplification of Terms**:
Simplify the expression \(\left(\frac{5n-1}{7n+2}\right)^n\) for large \(n\):
\[
\frac{5n-1}{7n+2} \approx \frac{5n}{7n} = \frac{5}{7}
\]
3. **Apply the Root Test**:
The Root Test involves taking the \(n\)-th root of \(a_n^n\):
\[
\sqrt[n]{\left(\frac{5n-1}{7n+2}\right)^n} = \left(\frac{5n-1}{7n+2}\right)
\]
4. **Limit Analysis**:
Determine the limit as \(n\) approaches infinity:
\[
\lim_{n \to \infty} \left(\frac{5n-1}{7n+2}\right) = \frac{5}{7}
\]
If this limit is less than 1, the Root Test would indicate that the
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