Does the following series converge or diverge? 2n Σ n2 n= 1 +2 O The series diverges. The series converges.
Does the following series converge or diverge? 2n Σ n2 n= 1 +2 O The series diverges. The series converges.
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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![### Does the following series converge or diverge?
\[ \sum_{n=1}^{\infty} \frac{2n}{n^2 + 2} \]
- ○ The series diverges.
- ○ The series converges.
---
**Explanation:**
The given problem asks whether the infinite series converges or diverges. The series is defined as the sum of terms in the form:
\[ \frac{2n}{n^2 + 2} \]
Each term in the series is a fraction where the numerator is \( 2n \) and the denominator is \( n^2 + 2 \). To solve this, we would need to apply convergence tests such as the Limit Comparison Test or the Ratio Test, among others, to determine the behavior of the series as \( n \) approaches infinity.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1e4fcbe0-ed74-4447-b38a-a2af13f9fabc%2F2b58822a-f217-4bf6-93d5-a68e388a2133%2F90bfcfk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Does the following series converge or diverge?
\[ \sum_{n=1}^{\infty} \frac{2n}{n^2 + 2} \]
- ○ The series diverges.
- ○ The series converges.
---
**Explanation:**
The given problem asks whether the infinite series converges or diverges. The series is defined as the sum of terms in the form:
\[ \frac{2n}{n^2 + 2} \]
Each term in the series is a fraction where the numerator is \( 2n \) and the denominator is \( n^2 + 2 \). To solve this, we would need to apply convergence tests such as the Limit Comparison Test or the Ratio Test, among others, to determine the behavior of the series as \( n \) approaches infinity.
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