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(b) Show that the sequence defined by a₁ = 3 and for n > 1, an = √3+an-1 converges. Find the limit. Hint: Show the sequence is increasing, and use induction to show an ≤ 3, VnEJ. Hind Winte (S. D - hint L

(b) Show that the sequence defined by a₁ = 3 and for n > 1, an = √3+an-1 converges. Find the limit. Hint: Show the sequence is increasing, and use induction to show an ≤ 3, VnEJ. Hind Winte (S. D - hint L

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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Can you do 1b please? For the hint it's suppose to say "decreasing" .. thanks

1. (a) A sequence \(\{a_n\}_{n=1}^{\infty}\) is said to be increasing if each term is at least the previous term: that is, \(a_{n+1} \geq a_n\) \(\forall n \in J\). Show that an increasing sequence that is bounded from above converges. **Hint:** Think about how to define a point in \(R\) that will serve as the limit. Then use the definition of convergence. (b) Show that the sequence defined by \(a_1 = 3\) and for \(n > 1\), \(a_n = \sqrt{3 + a_{n-1}}\) converges. Find the limit. **Hint:** Show the sequence is increasing, and use induction to show \(a_n \leq 3\), \(\forall n \in J\). **HintHint:** (See Bernardo's announcement) Turns out hint is not entirely correct. The sequence is \(3, \sqrt{3 + \sqrt{3}}, \sqrt{3 + \sqrt{3 + \sqrt{3}}}, \ldots\).
Transcribed Image Text:1. (a) A sequence \(\{a_n\}_{n=1}^{\infty}\) is said to be increasing if each term is at least the previous term: that is, \(a_{n+1} \geq a_n\) \(\forall n \in J\). Show that an increasing sequence that is bounded from above converges. **Hint:** Think about how to define a point in \(R\) that will serve as the limit. Then use the definition of convergence. (b) Show that the sequence defined by \(a_1 = 3\) and for \(n > 1\), \(a_n = \sqrt{3 + a_{n-1}}\) converges. Find the limit. **Hint:** Show the sequence is increasing, and use induction to show \(a_n \leq 3\), \(\forall n \in J\). **HintHint:** (See Bernardo's announcement) Turns out hint is not entirely correct. The sequence is \(3, \sqrt{3 + \sqrt{3}}, \sqrt{3 + \sqrt{3 + \sqrt{3}}}, \ldots\).
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