What changes when a power series is differentiated?
You will be able to: Differentiate term by term inside the radius and recheck endpoints.
What changes when a power series is differentiated?
A position formula and its velocity formula describe related information. Differentiating a series similarly creates a related function, but its boundary behavior may change.
A useful starting point: How can substitution create a new power series? →
Words and symbols before equations
- Termwise differentiation
- Differentiate each power-series term.
- Same radius theorem
- A power series and its derivative have the same radius of convergence.
- Endpoint retest
- A new check after the terms change.
- Reindexing
- Renaming the index without changing which terms are added.
What this picture assumes
Controls stay strictly inside radius 1, where termwise rules hold. Substitution and derivative examples exclude both endpoints. The integrated arctangent series converges at both endpoints; equality there needs a limiting argument.
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.
- Substitute: 1/(1+x²). At x=0.5, 4 nonzero terms give 0.796875. Function value: 0.8. The termwise rule holds inside abs(x)<1; endpoint tests remain separate.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
Inside the radius, d/dx[Σcₙ(x−a)ⁿ]=Σₙ₌₁∞ncₙ(x−a)ⁿ⁻¹. The constant term differentiates to zero, so the derivative sum begins at n=1.
Differentiating 1/(1−x)=Σₙ₌₀∞xⁿ gives 1/(1−x)²=Σₙ₌₁∞nxⁿ⁻¹ for abs(x)<1. The radius remains 1.
The radius theorem does not preserve endpoint inclusion. For example, differentiating Σₙ₌₁∞xⁿ/n changes it to Σₙ₌₁∞xⁿ⁻¹, which loses its convergent endpoint x=−1.
A worked example, step by step
Differentiate the series for 1/(1−x), then multiply by x to represent x/(1−x)².
- The original series is 1+x+x²+… for abs(x)<1.
- Differentiate to get 1+2x+3x²+… .
- Multiply by x: x+2x²+3x³+…=Σₙ₌₁∞nxⁿ.
- At ±1 the terms fail to tend to zero, so the interval remains (−1,1).
The radius stays the same under differentiation; the full interval may not.
Why does the derivative sum often start at n=1?
Compare with an explanation
The n=0 constant term differentiates to zero.
Predict. Change one thing. Explain.
Select the derivative case. Compare the coefficients n+1 with the original geometric coefficients and check how the displayed partial polynomial approaches the derivative function.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Substitute: 1/(1+x²). At x=0.5, 4 nonzero terms give 0.796875. Function value: 0.8. The termwise rule holds inside abs(x)<1; endpoint tests remain separate.
Controls stay strictly inside radius 1, where termwise rules hold. Substitution and derivative examples exclude both endpoints. The integrated arctangent series converges at both endpoints; equality there needs a limiting argument.
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 questionDifferentiate F(x)=Σₙ₌₁∞xⁿ/n inside abs(x)<1. Test the derivative series at x=−1 and x=1.
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Compare with the answer and four-point rubric
- 1 point: F′(x)=Σₙ₌₁∞xⁿ⁻¹.
- 1 point: It is geometric and equals 1/(1−x) inside abs(x)<1.
- 1 point: At x=1 the terms are all 1, so it diverges.
- 1 point: At x=−1 terms alternate ±1 and do not tend to zero, so that endpoint also diverges.
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 happens to the radius?
It stays the same.
RECALL 2What happens to the endpoints?
They require fresh tests.
RECALL 3What happens to a constant term?
Its derivative is zero.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
What changes when a power series is differentiated?
- Differentiate termwise only inside the convergence interval’s interior, unless separately justified.
- Derivative series has the same radius.
- Always retest endpoints.
Remember: The radius stays the same under differentiation; the full interval may not.
Conditions: Controls stay strictly inside radius 1, where termwise rules hold. Substitution and derivative examples exclude both endpoints. The integrated arctangent series converges at both endpoints; equality there needs a limiting argument.
Refresh Kid · AP Calculus BC Unit 10 · Objectives LIM-8.G · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 10.15, LIM-8.G. 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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