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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![### Topic: Ratio Test for Series Convergence
#### Consider the Series
\[ \sum_{n=1}^{\infty} \frac{3^n n!}{5^n n^2 (n+1)!} \]
#### Instructions
In your written work:
1. **Perform the ratio test.**
2. **Show all work.**
#### Goal
Determine if the ratio test is:
- Inconclusive
- Shows that the series is convergent
- Shows that the series is divergent
#### Options
- ⬜ Convergent
- ⬜ Divergent
- ⬜ Inconclusive
---
### Explanation
The Ratio Test for the convergence of a series \(\sum a_n\) is defined as follows:
1. Calculate the limit:
\[ L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \]
2. Interpret the result with the following criteria:
- If \( L < 1 \), the series is **absolutely convergent**.
- If \( L > 1 \) or \( L = \infty \), the series is **divergent**.
- If \( L = 1 \), the test is **inconclusive**.
Apply the Ratio Test to the given series \(\sum_{n=1}^{\infty} \frac{3^n n!}{5^n n^2 (n+1)!}\) to determine its convergence properties.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd7b1f94a-aeb1-4e21-a353-b14674dcbf02%2F95f1de87-b8d0-49a2-9f79-79e57125db5e%2F4hczp7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Topic: Ratio Test for Series Convergence
#### Consider the Series
\[ \sum_{n=1}^{\infty} \frac{3^n n!}{5^n n^2 (n+1)!} \]
#### Instructions
In your written work:
1. **Perform the ratio test.**
2. **Show all work.**
#### Goal
Determine if the ratio test is:
- Inconclusive
- Shows that the series is convergent
- Shows that the series is divergent
#### Options
- ⬜ Convergent
- ⬜ Divergent
- ⬜ Inconclusive
---
### Explanation
The Ratio Test for the convergence of a series \(\sum a_n\) is defined as follows:
1. Calculate the limit:
\[ L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \]
2. Interpret the result with the following criteria:
- If \( L < 1 \), the series is **absolutely convergent**.
- If \( L > 1 \) or \( L = \infty \), the series is **divergent**.
- If \( L = 1 \), the test is **inconclusive**.
Apply the Ratio Test to the given series \(\sum_{n=1}^{\infty} \frac{3^n n!}{5^n n^2 (n+1)!}\) to determine its convergence properties.
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