3" n! Σ n=1 5"n2(n+ 1)!

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...
icon
Related questions
Question
### 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.
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.
Expert Solution
steps

Step by step

Solved in 2 steps with 1 images

Blurred answer
Knowledge Booster
Multivariate Distributions and Functions of Random Variables
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Precalculus
Precalculus
Calculus
ISBN:
9780135189405
Author:
Michael Sullivan
Publisher:
PEARSON
Calculus: Early Transcendental Functions
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning