Exercise · ZW-MONOTONE-CONVERGENCE-THEOREM-010 · ★★★ Advanced
Let \(a_n=\sum_{k=0}^{n}\frac1{k!}\), increasing with \(a_n<3\) for all \(n\). What follows?
Choose one option.
Show answer
- A. It converges to some \(L\) with \(\tfrac52\le L\le3\)
Explanation
It is monotone and bounded, so it converges. Being increasing, \(L\ge a_2=1+1+\tfrac12=\tfrac52\), and by the bound \(L\le3\). (In fact \(L=e\approx2.718\), but the theorem already gives existence.)
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: