Calculus II · Unit 4A · lesson
Monotone and Bounded Sequences
Use monotonicity, bounds, induction, and the Monotone Convergence Theorem to analyze recursively defined sequences before solving for a limit.
Section overview
Sequences and their limitsWhat this section is building
Use monotonicity, bounds, induction, and the Monotone Convergence Theorem to analyze recursively defined sequences before solving for a limit.
A sequence converges when every sufficiently late term remains arbitrarily close to one finite target.
Use algebraic limits when a formula is explicit; use bounds and monotonicity when a recurrence hides the formula.
Reading a finite plot as proof or solving a recurrence's fixed-point equation before proving convergence.
Learning objectives
identify monotonicity and bounds and use the Monotone Convergence Theorem.
Monotone and Bounded Sequences
A sequence can be trapped by its own direction of motion
A monotone sequence moves in only one direction: never down, or never up. Direction alone is not enough, because increases forever. A bound alone is not enough, because stays between and while oscillating. Together, monotonicity and boundedness prevent both escape and endless backtracking.
The Monotone Convergence Theorem is an existence theorem. It can prove that a limit exists before we know its exact value. This is particularly useful for recursively defined sequences, where algebraic formulas may be unavailable. Once existence is established, a recurrence can sometimes be passed to the limit to identify possible values, after which bounds select the valid one.
Direction and confinement solve different problems
A sequence can increase forever and escape to infinity, or remain bounded while oscillating. Monotonicity prevents reversal; boundedness prevents escape. Together they force convergence.
For recursive sequences, prove these properties before solving a fixed-point equation. The usual pattern is induction for the bound, an inequality for monotonicity, the Monotone Convergence Theorem, and only then the equation for the limiting value.
The supremum supplies the target
For an increasing bounded sequence, let be its least upper bound. Since is not an upper bound, some term exceeds it; every later term remains between and . Hence the sequence converges to .
Read this graph as text
Monotone plus bounded forces convergence. The increasing sequence a n=1-1/n can never cross its upper bound 1, so its terms are squeezed into a narrowing vertical range. Increasing staircase beneath a least upper bound. Preserve the misconception control described in the lesson and storyboard.
Written labels, distinct line styles, markers, and fill patterns communicate every relationship in monotone plus bounded forces convergence; color is never the only cue.
Why it matters: Increasing staircase beneath a least upper bound.
The increasing sequence can never cross its upper bound , so its terms are squeezed into a narrowing vertical range.
Monotone plus bounded forces convergence. Increasing staircase beneath a least upper bound.
Each term is at least the preceding term, while the dashed line blocks upward escape. The theorem guarantees a finite limit even before its exact value is calculated.
Monotone Convergence Theorem
Every increasing sequence that is bounded above converges. Every decreasing sequence that is bounded below converges.
Recursive-sequence proof order
For a recursive sequence, establish a bound, establish monotonicity, invoke the theorem to prove a limit exists, and only then pass to the limit in the recurrence. Solving the fixed-point equation first finds candidates; it does not prove that the sequence reaches any of them.
A recursive sequence
Let and . One can show inductively that and that . Therefore the sequence converges. If , then
so . The bounds force , not .
Prove convergence before solving for the limit
Let and . If , then , so induction gives an upper bound. For ,
Thus the sequence is increasing and bounded, so it converges. Its limit satisfies , and the admissible solution is .
Suprema hiding behind the theorem
A rigorous proof uses the least-upper-bound property of the real numbers. An increasing bounded sequence has a supremum ; if the terms failed to approach , then some number below would still be an upper bound, contradicting leastness. This is one of the first places where completeness of does real work.
A fixed-point equation does not prove convergence
Solving identifies possible limits only. Establish monotonicity and boundedness first.
u4a-monotone_and_bounded_sequences-01Find .
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Show hint
The sequence is increasing and bounded above by ; direct algebra also reveals the value.
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Show that is increasing and bounded above.
Determine whether is monotone. Does it still converge?
For , explain why a positive limit must equal .
State exactly what monotonicity contributes and what boundedness contributes.
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