How does a power series define a radius of convergence?
You will be able to: Find an open convergence interval from the ratio test.
How does a power series define a radius of convergence?
A polynomial accepts any real input, but an infinite polynomial may only work near its center. The distance from that center can decide convergence.
A useful starting point: How do you choose enough Taylor terms for a tolerance? →
Words and symbols before equations
- Power series
- Σcₙ(x−a)ⁿ with constant coefficients cₙ.
- Center a
- The input where all positive-power terms vanish.
- Radius R
- Distance from the center within which absolute convergence holds.
- Open interval
- a−R<x<a+R, excluding endpoints for separate testing.
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
Apply a series test while treating x as fixed. For Σₙ₌₁∞(x−2)ⁿ/(n3ⁿ), the absolute consecutive ratio tends to abs(x−2)/3.
The ratio test gives absolute convergence when abs(x−2)<3, or −1<x<5, and divergence outside. At x=2 every term is zero, so it converges directly even though the ratio of zero terms is undefined.
The radius is R=3. The ratio limit equals 1 at the endpoints, so they need their own tests before the full interval is known.
A worked example, step by step
Find the radius and open convergence interval for Σₙ₌₀∞(x+1)ⁿ/4ⁿ.
- This is geometric with ratio (x+1)/4.
- Convergence requires abs((x+1)/4)<1.
- Thus abs(x+1)<4, or −5<x<3.
- The center is −1 and radius is 4; endpoint tests are still needed for a complete interval.
The radius is a nonnegative distance, not an interval or the right endpoint.
Does the ratio test decide endpoints when its limit is 1?
Compare with an explanation
No. Substitute each endpoint into the original series.
Predict. Change one thing. Explain.
Move x across −1, 2 and 5 in the model. Compare finite partial sums with the exact classification; identify the center, radius and unresolved ratio-test endpoints.
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 questionFor Σₙ₌₁∞(x−4)ⁿ/(n2ⁿ), use the ratio test to find R and the open interval.
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Compare with the answer and four-point rubric
- 1 point: The absolute consecutive ratio is abs(x−4)n/[2(n+1)].
- 1 point: Its limit is abs(x−4)/2.
- 1 point: Require abs(x−4)<2.
- 1 point: R=2 and the open interval is (2,6); endpoints remain separate.
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 is the radius?
The distance from the center to the boundary of the convergence region.
RECALL 2What does the ratio inequality give first?
An open interval of absolute convergence.
RECALL 3Why inspect the center separately?
The original series is defined there even when a term ratio is 0/0.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How does a power series define a radius of convergence?
- A power series converges absolutely for abs(x−a)<R.
- It diverges for abs(x−a)>R.
- Check x=a directly if a ratio expression divides zero by zero.
Remember: The radius is a nonnegative distance, not an interval or the right endpoint.
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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