4. Determine whether the following series is convergent or divergent. 1.3.5... (2n-1) 2.4.6...(2n) (3+1) (√7-1)") n=1
4. Determine whether the following series is convergent or divergent. 1.3.5... (2n-1) 2.4.6...(2n) (3+1) (√7-1)") n=1
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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![**Problem Statement:**
Determine whether the following series is convergent or divergent.
\[
\sum_{n=1}^{\infty} \left( \frac{1 \cdot 3 \cdot 5 \cdot \ldots \cdot (2n-1)}{2 \cdot 4 \cdot 6 \cdot \ldots \cdot (2n)}(3^n + 1) + (\sqrt[7]{7} - 1)^n \right)
\]
**Explanation:**
The given series involves two main components:
1. A ratio of products of odd and even numbers:
- The numerator is a product of odd numbers from 1 to \( (2n - 1) \).
- The denominator is a product of even numbers from 2 to \( 2n \).
2. An expression involving \( 3^n + 1 \).
3. An exponential term \((\sqrt[7]{7} - 1)^n\).
To determine if the series is convergent or divergent, the relevant tests for convergence, such as the Ratio Test or Comparison Test, may be applied based on the behavior of the terms as \( n \) approaches infinity.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd789b8b1-9579-4fbd-b192-1d0f61eccfe1%2F312e6268-bad3-4072-a690-f5bc269487fb%2Fomztg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Determine whether the following series is convergent or divergent.
\[
\sum_{n=1}^{\infty} \left( \frac{1 \cdot 3 \cdot 5 \cdot \ldots \cdot (2n-1)}{2 \cdot 4 \cdot 6 \cdot \ldots \cdot (2n)}(3^n + 1) + (\sqrt[7]{7} - 1)^n \right)
\]
**Explanation:**
The given series involves two main components:
1. A ratio of products of odd and even numbers:
- The numerator is a product of odd numbers from 1 to \( (2n - 1) \).
- The denominator is a product of even numbers from 2 to \( 2n \).
2. An expression involving \( 3^n + 1 \).
3. An exponential term \((\sqrt[7]{7} - 1)^n\).
To determine if the series is convergent or divergent, the relevant tests for convergence, such as the Ratio Test or Comparison Test, may be applied based on the behavior of the terms as \( n \) approaches infinity.
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