Does sin () x 7 converge or diverge? Why?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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![## Convergence and Divergence of Series
### Problem Statement
Does
\[
\sum_{n=1}^{\infty} \sin\left(\frac{\pi n}{2}\right) x^{-\frac{1}{2}}
\]
converge or diverge? Why?
### Explanation
This mathematical expression asks whether the infinite series, defined by the terms \(\sin\left(\frac{\pi n}{2}\right) x^{-\frac{1}{2}}\) for all \(n\) starting from 1 to infinity, converges to a finite value or diverges. Understanding convergence or divergence involves analyzing the behavior of this series as \(n\) approaches infinity.
Key points to consider when analyzing the convergence of this series include:
- The behavior of the sine function in the terms \(\sin\left(\frac{\pi n}{2}\right)\).
- The impact of the power of \(x^{-1/2}\).
- The combined effect of these factors on the infinite summation.
This problem requires a foundational understanding of series analysis, including tests of convergence such as the comparison test, ratio test, or alternating series test, depending on the structure of the series.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc247148b-ac88-4d23-ab16-d846cf9acaa7%2F668b9f07-c995-413c-ad39-f7e37754c978%2Ffwj86mw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Convergence and Divergence of Series
### Problem Statement
Does
\[
\sum_{n=1}^{\infty} \sin\left(\frac{\pi n}{2}\right) x^{-\frac{1}{2}}
\]
converge or diverge? Why?
### Explanation
This mathematical expression asks whether the infinite series, defined by the terms \(\sin\left(\frac{\pi n}{2}\right) x^{-\frac{1}{2}}\) for all \(n\) starting from 1 to infinity, converges to a finite value or diverges. Understanding convergence or divergence involves analyzing the behavior of this series as \(n\) approaches infinity.
Key points to consider when analyzing the convergence of this series include:
- The behavior of the sine function in the terms \(\sin\left(\frac{\pi n}{2}\right)\).
- The impact of the power of \(x^{-1/2}\).
- The combined effect of these factors on the infinite summation.
This problem requires a foundational understanding of series analysis, including tests of convergence such as the comparison test, ratio test, or alternating series test, depending on the structure of the series.
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