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3) Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? a) an = 1-n 2+n b) an = 2 + (-1)" n

3) Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? a) an = 1-n 2+n b) an = 2 + (-1)" n

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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**Question 3: Sequence Analysis** Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? **a)** \( a_n = \frac{1-n}{2+n} \) **b)** \( a_n = 2 + \frac{(-1)^n}{n} \) **Explanation for a):** - The sequence is defined by the formula \( a_n = \frac{1-n}{2+n} \). - To determine if the sequence is increasing or decreasing, analyze the sign of \( a_{n+1} - a_n \). **Explanation for b):** - The sequence is defined by the formula \( a_n = 2 + \frac{(-1)^n}{n} \). - This introduces an alternating component \( \frac{(-1)^n}{n} \) that needs to be considered to determine if the sequence is monotonic. **Boundedness:** - For both sequences, analyze the limiting behavior as \( n \to \infty \) to determine boundedness.
Transcribed Image Text:**Question 3: Sequence Analysis** Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? **a)** \( a_n = \frac{1-n}{2+n} \) **b)** \( a_n = 2 + \frac{(-1)^n}{n} \) **Explanation for a):** - The sequence is defined by the formula \( a_n = \frac{1-n}{2+n} \). - To determine if the sequence is increasing or decreasing, analyze the sign of \( a_{n+1} - a_n \). **Explanation for b):** - The sequence is defined by the formula \( a_n = 2 + \frac{(-1)^n}{n} \). - This introduces an alternating component \( \frac{(-1)^n}{n} \) that needs to be considered to determine if the sequence is monotonic. **Boundedness:** - For both sequences, analyze the limiting behavior as \( n \to \infty \) to determine boundedness.
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