Test the series below for convergence using the Ratio Test. n + 5 33n + 2 T=1 The limit of the ratio test simplifies to lim f(n)| where /(n) The limit is: (enter oo for infinity if needed)
Test the series below for convergence using the Ratio Test. n + 5 33n + 2 T=1 The limit of the ratio test simplifies to lim f(n)| where /(n) The limit is: (enter oo for infinity if needed)
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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![**Testing Series Convergence Using the Ratio Test**
The following series is presented for testing convergence:
\[ \sum_{n=1}^{\infty} \frac{n + 5}{3^{3n+2}} \]
### Steps to Find Convergence:
1. **Ratio Test Setup**: Identify the series term and apply the ratio test.
2. **Compute the Ratio**:
\[ \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \]
3. **Compare to 1**: If the limit is less than 1, the series converges; if greater than 1, it diverges; if equal to 1, the test is inconclusive.
### Determine \( f(n) \) and Calculate the Limit
#### Expressions:
- \( f(n) = \) [Enter your expression for the function]
- The limit is: [Enter calculated limit]
*(enter \( \infty \) for infinity if needed)*
This framework outlines how to approach the convergence of a given series through the ratio test methodology, using the specific series provided above.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F09b3997c-2096-4573-b74b-055c69181c0d%2F34984591-1a79-4ec7-afbc-830e8761a72c%2Fxcry8s_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Testing Series Convergence Using the Ratio Test**
The following series is presented for testing convergence:
\[ \sum_{n=1}^{\infty} \frac{n + 5}{3^{3n+2}} \]
### Steps to Find Convergence:
1. **Ratio Test Setup**: Identify the series term and apply the ratio test.
2. **Compute the Ratio**:
\[ \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \]
3. **Compare to 1**: If the limit is less than 1, the series converges; if greater than 1, it diverges; if equal to 1, the test is inconclusive.
### Determine \( f(n) \) and Calculate the Limit
#### Expressions:
- \( f(n) = \) [Enter your expression for the function]
- The limit is: [Enter calculated limit]
*(enter \( \infty \) for infinity if needed)*
This framework outlines how to approach the convergence of a given series through the ratio test methodology, using the specific series provided above.
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