n=2 (Inn)² n³ 3
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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Question
determine whether the series converges absolutely, converges conditionally, or diverges. state the test used to draw your conclusions
![The image presents a mathematical series, expressed as follows:
\[
\sum_{n=2}^{\infty} \frac{2 (\ln n)^2}{n^3}
\]
### Explanation:
- **Summation (\(\sum\))**: The notation indicates the sum of terms in a sequence. The summation begins at \(n = 2\) and continues to infinity (\(\infty\)).
- **\((\ln n)^2\)**: This represents the natural logarithm of \(n\), squared. The natural logarithm is the logarithm to the base \(e\), where \(e\) is approximately equal to 2.718.
- **\(n^3\)**: This term indicates that each \(n\) is raised to the power of 3.
- **Coefficient 2**: This scalar multiplies the entire fraction \(\frac{(\ln n)^2}{n^3}\).
This series likely appears in discussions of convergence or divergence in the context of mathematical analysis, particularly in calculus or advanced calculus courses. The behavior of such series, especially in terms of convergence, may be explored to develop a deeper understanding of mathematical functions and sequences.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6a85ba4d-6f5c-41bf-97e5-182adb23b838%2Fb20d4ee2-0f4c-4939-9f0e-3386e8cf3d88%2Fd0uugb_processed.png&w=3840&q=75)
Transcribed Image Text:The image presents a mathematical series, expressed as follows:
\[
\sum_{n=2}^{\infty} \frac{2 (\ln n)^2}{n^3}
\]
### Explanation:
- **Summation (\(\sum\))**: The notation indicates the sum of terms in a sequence. The summation begins at \(n = 2\) and continues to infinity (\(\infty\)).
- **\((\ln n)^2\)**: This represents the natural logarithm of \(n\), squared. The natural logarithm is the logarithm to the base \(e\), where \(e\) is approximately equal to 2.718.
- **\(n^3\)**: This term indicates that each \(n\) is raised to the power of 3.
- **Coefficient 2**: This scalar multiplies the entire fraction \(\frac{(\ln n)^2}{n^3}\).
This series likely appears in discussions of convergence or divergence in the context of mathematical analysis, particularly in calculus or advanced calculus courses. The behavior of such series, especially in terms of convergence, may be explored to develop a deeper understanding of mathematical functions and sequences.
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