Exercise · ZW-MONOTONE-CONVERGENCE-THEOREM-005 · ★☆☆ Basic
Let \(a_1=\sqrt2\) and \(a_{n+1}=\sqrt{2+a_n}\), increasing and bounded by 2. If it converges to \(L\), what is \(L\)?
Choose one option.
Show answer
- B. 2
Explanation
Taking limits in the recurrence, \(L=\sqrt{2+L}\), so \(L^2=L+2\). Its solutions are \(2\) and \(-1\); since the terms are positive, \(L=2\).
Review the concept: Monotone convergence theorem
Every real sequence \((a_n)\), indexed by the natural numbers \(\mathbb N\), that is monotone and bounded converges to a real number.
\[a_n\ \text{increasing and bounded}\ \Rightarrow\ \lim a_n=\sup\{a_n:n\in\mathbb N\}\]
More practice problems: monotone convergence theorem
- ZW-MONOTONE-CONVERGENCE-THEOREM-001 The sequence a_n=1-1n is increasing and bounded above by 1. What does it converge to?
- ZW-MONOTONE-CONVERGENCE-THEOREM-002 The sequence a_n=1n^2 is decreasing and positive. What does it converge to?
- ZW-MONOTONE-CONVERGENCE-THEOREM-003 The sequence a_n=n is increasing but does not converge. Which hypothesis of the theorem fails?
- ZW-MONOTONE-CONVERGENCE-THEOREM-004 The sequence a_n=(-1)^n is bounded and does not converge. Why does this not contradict the theorem?
- ZW-MONOTONE-CONVERGENCE-THEOREM-006 Let a_1=0 and a_n+1=(a_n)/(3)+2, increasing and bounded by 3. What is its limit?
- ZW-MONOTONE-CONVERGENCE-THEOREM-007 An increasing sequence satisfies a_n≤7 for all n and converges to L. What follows?
- ZW-MONOTONE-CONVERGENCE-THEOREM-008 A decreasing sequence that is bounded below converges to:
- ZW-MONOTONE-CONVERGENCE-THEOREM-009 Which of these statements is FALSE?
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