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Determine whether each of the following Sequences is convergent. Justify!! @@ (x₁) where x := n 16. с 142 d^+2 n c't dn ← Sin (nπ/6) where окска (X₁) where x ₁² = (-1)^ ()

Determine whether each of the following Sequences is convergent. Justify!! @@ (x₁) where x := n 16. с 142 d^+2 n c't dn ← Sin (nπ/6) where окска (X₁) where x ₁² = (-1)^ ()

Advanced Engineering Mathematics
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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C
**Title: Convergence of Sequences** **Introduction:** In this exercise, we will determine whether each given sequence is convergent and provide a justification for our conclusion. **Sequence A:** \[ (x_n) \text{ where } x_n = \frac{\sin(n\pi/6)}{n} \] **Sequence B:** \[ \left( \frac{c^{n+2} - d^{n+2}}{c^n + d^n} \right) \text{ where } 0 < c < d \] **Sequence C:** \[ (x_n) \text{ where } x_n = (-1)^n \left(\frac{n-1}{n}\right) \] **Explanation:** You will explore each sequence, analyze their behavior as \( n \) approaches infinity, and determine whether they converge to a specific limit. Consider any known theorems or properties of limits that can aid in the justification of your conclusions.
Transcribed Image Text:**Title: Convergence of Sequences** **Introduction:** In this exercise, we will determine whether each given sequence is convergent and provide a justification for our conclusion. **Sequence A:** \[ (x_n) \text{ where } x_n = \frac{\sin(n\pi/6)}{n} \] **Sequence B:** \[ \left( \frac{c^{n+2} - d^{n+2}}{c^n + d^n} \right) \text{ where } 0 < c < d \] **Sequence C:** \[ (x_n) \text{ where } x_n = (-1)^n \left(\frac{n-1}{n}\right) \] **Explanation:** You will explore each sequence, analyze their behavior as \( n \) approaches infinity, and determine whether they converge to a specific limit. Consider any known theorems or properties of limits that can aid in the justification of your conclusions.
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Step 1: Question Description

Determine whether each of the following sequences is convergent. Justify.


C. open parentheses x subscript n close parentheses  where x subscript n equals open parentheses negative 1 close parentheses to the power of n open parentheses fraction numerator n minus 1 over denominator n end fraction close parentheses.

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