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Math101learn.math101.caMonotone Sequences
A rigorous, example-driven guide to monotone sequences, including hypotheses, method choice, verification, and practice.
The central idea
A sequence is increasing if $a_{n+1}\ge a_n$ and decreasing if $a_{n+1}\le a_n$. It is monotone if one relation holds throughout or eventually. The Monotone Convergence Theorem states that an increasing sequence bounded above, or a decreasing sequence bounded below, converges.
Definitions, hypotheses, and notation
Eventually monotone behavior is enough for convergence because deleting a finite prefix does not affect the limit. Strict inequalities are unnecessary; repeated equal terms are compatible with monotonicity. For recursive sequences, a bound often ensures the recurrence stays in a domain where its update function preserves order.
The theorem proves existence of a limit without initially knowing its value. Once convergence is established, continuity of the recurrence function can justify passing to the limit. Multiple fixed points require initial conditions and bounds to identify the attainable one.
Conceptual meaning
Monotonicity prevents back-and-forth oscillation; boundedness prevents escape to infinity. Together they force terms toward the least upper bound or greatest lower bound. Either condition alone is insufficient.
A dependable method and decision rule
- Compute $a_{n+1}-a_n$ or $a_{n+1}/a_n$ when terms are positive.
- Establish one sign for the difference or compare the ratio with one.
- Find an explicit upper or lower bound.
- Invoke monotone convergence only after both properties are proved.
- If a recurrence is given, solve the candidate fixed-point equation and justify which root is reached.
Fully worked example
Graphical or geometric meaning
On a number line the terms march rightward without crossing the barrier at one, and the gaps shrink. A plot of $(n,a_n)$ shows discrete points approaching the horizontal line $y=1$.
Common mistakes and why they fail
Verification and reasonableness checks
- Prove the bound for every relevant index, often by induction.
- Compare with a direct limit when a closed formula exists.
- Test whether an alleged monotone direction matches early terms.
Monotonicity needs a bound
A nondecreasing sequence bounded above converges to its supremum; a nonincreasing sequence bounded below converges to its infimum. Monotonicity alone permits escape to infinity, while boundedness alone permits oscillation. For a recursive sequence, prove both properties—often by induction—before taking limits in the recurrence. If convergence to $L$ has been established and the recurrence is continuous at relevant values, $L$ must satisfy a fixed-point equation. That equation may have several solutions, so use bounds and the initial term to select the possible one. Solving for fixed points first identifies candidates but does not prove that the sequence approaches any of them. Keep those logical steps in the right order.
Practice
- Is $a_n=n/(n+1)$ increasing?
- Is $(-1)^n$ bounded? convergent?
- Is $a_n=n$ monotone? convergent?
Answers and brief solutions
- Yes, and it tends to $1$.
- Bounded but not convergent.
- Increasing but divergent.
Connections and next steps
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Why must a_n=1−1/n converge?
- a_(n+1)−a_n=1/[n(n+1)]>0.
- Every term is less than 1.
- Increasing plus bounded above implies convergence.
End of lesson
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