How do you choose a test and justify the choice?
You will be able to: Select a test from the structure of a series and verify its conditions.
How do you choose a test and justify the choice?
Choosing a tool is easier after looking at the job. A fixed multiplier, a power of n, alternating signs and factorials each point toward different useful tests.
A useful starting point: How do factorials and exponentials suggest the ratio test? →
Words and symbols before equations
- Structure
- Features of the term formula that suggest a test.
- Benchmark
- A series with known behavior.
- Sufficient condition
- A condition that guarantees a result when satisfied.
- Inconclusive result
- A reason to try another method, not a final classification.
What this picture assumes
All series start at n=1. These examples demonstrate test selection, not an exhaustive algorithm. A theorem and its hypotheses accompany each conclusion.
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.
- Geometric series First term 1/4; common ratio 1/4. abs(r)<1, so it converges. Sum = (1/4)/(1−1/4) = 1/3.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
Start with the term limit. A nonzero or nonexistent limit settles divergence immediately. Next look for a geometric series or a p-series.
For positive rational expressions, compare dominant powers. For factorials or exponentials mixed with other factors, simplify a consecutive-term ratio. Alternating signs suggest checking absolute convergence and, if needed, the alternating series test.
A complete conclusion names the test and checks its hypotheses. The AP scope includes divergence, integral, comparison, alternating and ratio tests. The root test is not required here.
A worked example, step by step
Choose and apply a useful test for Σ(n+2)/(n³+1).
- Terms are positive and tend to zero, so the divergence test is inconclusive.
- Dominant powers suggest comparison with 1/n².
- The ratio to 1/n² is n²(n+2)/(n³+1)→1.
- Limit comparison with a convergent p-series proves convergence.
Do not list test names without conditions. A ratio limit of 1 means change methods.
What next if the ratio test gives L=1 for 1/n²?
Compare with an explanation
Recognize the p-series with p=2, which converges.
Predict. Change one thing. Explain.
Select each example. Before reading its decision, name a likely test and explain what condition you would verify. Compare your reasoning with the displayed justification.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Geometric series First term 1/4; common ratio 1/4. abs(r)<1, so it converges. Sum = (1/4)/(1−1/4) = 1/3.
All series start at n=1. These examples demonstrate test selection, not an exhaustive algorithm. A theorem and its hypotheses accompany each conclusion.
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 questionClassify Σ(1/4)ⁿ, Σ1/n and Σ2ⁿ/n! for n≥1, naming a suitable test for each.
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Compare with the answer and four-point rubric
- 1 point: The geometric ratio 1/4 has magnitude below 1, so the first converges.
- 1 point: The harmonic p-series has p=1 and diverges.
- 1 point: For the third, the consecutive ratio is 2/(n+1)→0.
- 1 point: The ratio test proves absolute convergence of the third.
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 should precede choosing a complicated test?
Check whether the terms tend to zero and whether a simple known form applies.
RECALL 2What must accompany a test name?
Verified hypotheses and a supporting calculation.
RECALL 3Is the root test required in this AP unit?
No.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How do you choose a test and justify the choice?
- Check the term limit first.
- Match a test to the algebraic structure.
- State conditions, computation and conclusion.
Remember: Do not list test names without conditions. A ratio limit of 1 means change methods.
Conditions: All series start at n=1. These examples demonstrate test selection, not an exhaustive algorithm. A theorem and its hypotheses accompany each conclusion.
Refresh Kid · AP Calculus BC Unit 10 · Objectives LIM-7.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 10.8, 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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