Why must each endpoint be tested separately?
You will be able to: Determine full convergence intervals, including radius zero and infinity cases.
Why must each endpoint be tested separately?
Two doors can sit the same distance from a room’s center yet have different locks. The two boundary inputs of a power series likewise can behave differently.
A useful starting point: How does a power series define a radius of convergence? →
Words and symbols before equations
- Interval of convergence
- All real inputs at which the power series converges.
- Closed endpoint
- An endpoint included because its substituted series converges.
- Radius zero
- Only the center converges.
- Infinite radius
- Every real input converges.
What this picture assumes
Σₙ₌₁∞(x−2)ⁿ/(n3ⁿ) has center 2 and radius 3. Its interval is [−1,5). Endpoint decisions come from alternating and harmonic tests, not from a finite graph.
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.
- x=3; S_N=0.405373 after N=6 terms. Ratio-limit expression abs(x−2)/3=0.333333. Absolutely convergent inside the radius. At x=2 all terms vanish; check directly instead of dividing zero by zero.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
For Σ(x−2)ⁿ/(n3ⁿ), at x=−1 the series becomes Σ(−1)ⁿ/n, conditionally convergent. At x=5 it becomes Σ1/n, divergent. The full interval is [−1,5).
Endpoints require fresh series tests, not just plugging into abs(x−a)<R. One, both or neither finite endpoint may be included.
For Σₙ₌₀∞xⁿ/n!, the ratio tends to 0 for every fixed x, so R=∞. For Σₙ₌₀∞n!xⁿ, ratios grow without bound for x≠0; only x=0 converges, so R=0.
A worked example, step by step
Find the interval for Σₙ₌₁∞(x+1)ⁿ/(n2ⁿ).
- The ratio test gives abs(x+1)<2, hence −3<x<1.
- At x=−3, the series is Σ(−1)ⁿ/n and converges by the alternating test.
- At x=1, it is harmonic and diverges.
- The interval is [−3,1), with R=2.
Do not copy endpoint behavior from a different series; the coefficients matter.
Can an endpoint be conditionally convergent?
Compare with an explanation
Yes; the left endpoint of this example is an alternating harmonic series.
Predict. Change one thing. Explain.
Inspect x=−1 and x=5 for the displayed power series. Explain the alternating and harmonic tests separately, even though both ratio limits are 1.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
x=3; S_N=0.405373 after N=6 terms. Ratio-limit expression abs(x−2)/3=0.333333. Absolutely convergent inside the radius. At x=2 all terms vanish; check directly instead of dividing zero by zero.
Σₙ₌₁∞(x−2)ⁿ/(n3ⁿ) has center 2 and radius 3. Its interval is [−1,5). Endpoint decisions come from alternating and harmonic tests, not from a finite graph.
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 questionDetermine the full interval for Σₙ₌₁∞xⁿ/n², explaining both endpoints.
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Compare with the answer and four-point rubric
- 1 point: The ratio limit is abs(x), so the open interval is (−1,1).
- 1 point: At x=1, Σ1/n² converges.
- 1 point: At x=−1, the absolute series is also Σ1/n² and converges.
- 1 point: The interval is [−1,1], and both endpoints converge absolutely.
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 1How do you decide endpoint inclusion?
Substitute and use a series test.
RECALL 2Can a radius be zero?
Yes; then only the center converges.
RECALL 3What does R=∞ mean?
Every real input is in the interval of convergence.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Why must each endpoint be tested separately?
- Substitute each endpoint into the original series.
- R=0: center only. R=∞: all real x.
- An interval can include one, both or neither finite endpoint.
Remember: Do not copy endpoint behavior from a different series; the coefficients matter.
Conditions: Σₙ₌₁∞(x−2)ⁿ/(n3ⁿ) has center 2 and radius 3. Its interval is [−1,5). Endpoint decisions come from alternating and harmonic tests, not from a finite graph.
Refresh Kid · AP Calculus BC Unit 10 · Objectives LIM-8.D · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 10.13, LIM-8.D. 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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