Which direction must a comparison inequality go?
You will be able to: Use positive upper and lower comparisons correctly.
Which direction must a comparison inequality go?
If every item in one bill costs no more than the matching item in a finite-total bill, the smaller bill also has a bounded total. A larger bill needs a different argument.
A useful starting point: Why can tiny terms still add up without bound? →
Words and symbols before equations
- Direct comparison
- An inequality between corresponding nonnegative terms.
- Upper bound
- A term at least as large as the target term.
- Lower bound
- A term no larger than the target term.
- Benchmark
- A familiar geometric or p-series used for comparison.
What this picture assumes
Both sequences are positive for n≥1. The first uses a convergent upper bound; the second a divergent lower bound. The inequalities hold for every n≥1.
Read the picture in three steps
- Read the axes and labels first. Identify what each symbol and line represents. Read the units and fixed conditions before comparing quantities.
- At n=6, aₙ=0.027027, bₙ=0.027778, aₙ/bₙ=0.972973. Target ≤ convergent 1/n²: convergent. Ratio tends to 1.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
If 0≤aₙ≤bₙ and Σbₙ converges, then Σaₙ converges. Its increasing partial sums are bounded above.
If 0≤bₙ≤aₙ and Σbₙ diverges, then Σaₙ diverges. The larger totals cannot stay finite.
Being smaller than a divergent series, or larger than a convergent series, gives no conclusion. Check the direction and the benchmark together; comparisons may begin on a tail.
A worked example, step by step
Prove Σ1/(n²+1) converges.
- All terms are positive.
- Since n²+1≥n², taking reciprocals gives 1/(n²+1)≤1/n².
- Σ1/n² converges as a p-series with p=2.
- The target converges by direct comparison with a convergent upper bound.
Taking reciprocals reverses an inequality between positive denominators.
Does aₙ≤1/n prove convergence?
Compare with an explanation
No; the upper benchmark is divergent.
Predict. Change one thing. Explain.
Compare 1/(n²+1) with 1/n². Then switch to 1/√(n²+1) versus 1/(√2 n). State why the second is a divergent lower bound for n≥1.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
At n=6, aₙ=0.027027, bₙ=0.027778, aₙ/bₙ=0.972973. Target ≤ convergent 1/n²: convergent. Ratio tends to 1.
Both sequences are positive for n≥1. The first uses a convergent upper bound; the second a divergent lower bound. The inequalities hold for every n≥1.
Explain what you noticed: Answer the investigation prompt above. State one observation and explain it using the term formula, partial sums, test conditions, remainder bound or interval of convergence. Identify what the representation cannot tell you.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionProve Σ1/(n+1) diverges by comparing it with 1/(2n), for n≥1.
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Compare with the answer and four-point rubric
- 1 point: n+1≤2n for n≥1.
- 1 point: Therefore 1/(n+1)≥1/(2n)>0.
- 1 point: Σ1/(2n) is half the divergent harmonic series.
- 1 point: The target diverges by direct comparison.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1What upper comparison proves convergence?
A convergent nonnegative series.
RECALL 2What lower comparison proves divergence?
A divergent nonnegative series.
RECALL 3Why are positivity and direction essential?
They allow ordered partial-sum bounds.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Which direction must a comparison inequality go?
- Smaller than a convergent positive series ⇒ convergent.
- Larger than a divergent positive series ⇒ divergent.
- The other two directions are inconclusive.
Remember: Taking reciprocals reverses an inequality between positive denominators.
Conditions: Both sequences are positive for n≥1. The first uses a convergent upper bound; the second a divergent lower bound. The inequalities hold for every n≥1.
Refresh Kid · AP Calculus BC Unit 10 · Objectives LIM-7.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 10.6, LIM-7.A. CED effective Fall 2020 and Fall 2026 clarifications checked September 17, 2026. This unit covers BC topics 10.1–10.15, Infinite Sequences and Series. Required convergence methods are the nth-term, integral, comparison, alternating and ratio tests; the root test is not assigned as required AP content.
Finite plots illustrate terms and partial sums but do not establish infinite convergence. Positive-series comparisons retain their hypotheses and inequality directions. Alternating bounds use the next omitted magnitude. Taylor bounds use the next derivative over the full interval. Power-series endpoints are checked separately, including after differentiation and integration. Taylor representation requires a remainder tending to zero.
All focused explanations, examples, practice and models are original Refresh Kid work. OpenStax was consulted for mathematical cross-checking. Khan Academy’s destination was checked, but JavaScript lesson content was not fully readable by the research tool. Organic Chemistry Tutor video titles, creator and destinations were checked; full videos were not reviewed. No provider questions, diagrams or scripts were copied. Resources are optional, and no paid resource is needed. Refresh Kid is not affiliated with or endorsed by these providers.
GitHub’s 3D website collection informed optional camera and spatial inspection. The original geometric slab model uses self-hosted Three.js with its MIT license. A unit cube receives slabs of widths 1/2, 1/4, 1/8 and so on, each spanning unit height and depth. The complete 2D strip and numerical volume readout remain available. No autoplay or WebGL is required to learn the mathematics.
Independent teacher review and observation of students remain pending. Technical checks do not certify mathematical accuracy, accessibility or learning effectiveness. This is a review edition.
Optional official resource: Released AP Calculus BC questions and scoring guides. The archive spans multiple units; no entire exam question is assigned here.
Learn → Explore → Practice → Review is informed by the IES learning guide; this implementation has not been evaluated with learners.
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