Exercise · ZW-MONOTONE-CONVERGENCE-THEOREM-009 · ★★★ Advanced
Which of these statements is FALSE?
Choose one option.
Show answer
- C. Every convergent sequence is monotone
Explanation
\(\frac{(-1)^n}{n}\) converges to 0 without being monotone. The other three are true: every convergent sequence is bounded (by the definition of limit); every monotone bounded sequence converges (by the theorem); and an increasing sequence unbounded above does not converge because every convergent sequence is bounded.
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-005 Let a_1=√2 and a_n+1=√2+a_n, increasing and bounded by 2. If it converges to L, what is L?
- 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:
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