## Convergence and Divergence of Series### Problem StatementDoes \[\sum_{n=1}^{\infty} \sin\left(\frac{\pi n}{2}\right) x^{-\frac{1}{2}}\]converge or diverge? Why? ### ExplanationThis 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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Answered: Does sin () x 7 converge or diverge?…

Math

Advanced Math

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

Section: Chapter Questions

Problem 1RQ

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Question

## 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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