USE THE GIVEN TEST TO DETERMINE THE CONVERGENCE DIVERGENCE OF THE GIVEN SERIES. P-SERIES TEST n+l n=0 n²³³ +n +2 b.) &
USE THE GIVEN TEST TO DETERMINE THE CONVERGENCE DIVERGENCE OF THE GIVEN SERIES. P-SERIES TEST n+l n=0 n²³³ +n +2 b.) &
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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![**Title: Determining Series Convergence Using the P-Series Test**
**Instructions:**
**Task:** Use the given test to determine the convergence or divergence of the given series.
**Method:** P-Series Test
**Series:**
\[
b.) \, \sum_{n=0}^{\infty} \frac{n+1}{3n^2 + n + 2}
\]
**Explanation:**
The P-Series Test is utilized to analyze the convergence of a series of the form \(\sum \frac{1}{n^p}\). In this problem, the series provided does not directly match the traditional form of a p-series, so further manipulation or comparison might be necessary before applying the test.
**Step-by-Step Guide:**
1. **Identify the series structure:** Determine if the series can be rewritten or compared to a standard p-series form.
2. **Check criteria for convergence:** If a series resembles a p-series, it converges if \(p > 1\) and diverges if \(p \leq 1\).
3. **Perform necessary comparisons or transformations:** Simplify or compare with a similar known convergent or divergent series if required.
Remember that detailed step-by-step calculation is necessary for academic comprehension.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0fef8a34-2f5f-4721-a6f5-64cfda708d6c%2F8a34c237-dbe8-43c2-b499-2ef3b9856446%2Fymjp9ru_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Determining Series Convergence Using the P-Series Test**
**Instructions:**
**Task:** Use the given test to determine the convergence or divergence of the given series.
**Method:** P-Series Test
**Series:**
\[
b.) \, \sum_{n=0}^{\infty} \frac{n+1}{3n^2 + n + 2}
\]
**Explanation:**
The P-Series Test is utilized to analyze the convergence of a series of the form \(\sum \frac{1}{n^p}\). In this problem, the series provided does not directly match the traditional form of a p-series, so further manipulation or comparison might be necessary before applying the test.
**Step-by-Step Guide:**
1. **Identify the series structure:** Determine if the series can be rewritten or compared to a standard p-series form.
2. **Check criteria for convergence:** If a series resembles a p-series, it converges if \(p > 1\) and diverges if \(p \leq 1\).
3. **Perform necessary comparisons or transformations:** Simplify or compare with a similar known convergent or divergent series if required.
Remember that detailed step-by-step calculation is necessary for academic comprehension.
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