How can substitution create a new power series?
You will be able to: Transform a known series and transform its convergence condition too.
How can substitution create a new power series?
A reusable recipe can make a new dish when one ingredient changes. A known series can likewise represent a new function after a carefully tracked substitution.
A useful starting point: When does a Taylor series actually equal its function? →
Words and symbols before equations
- Substitution
- Replacing the input of a known series with an expression.
- Composition
- A function applied to another expression.
- Coefficient scaling
- Multiplying each series coefficient by a constant.
- Transformed domain
- Inputs satisfying the original convergence restriction after substitution.
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
Start with 1/(1−u)=Σuⁿ for abs(u)<1. Substitute u=−x² to obtain 1/(1+x²)=Σ(−1)ⁿx²ⁿ for abs(x)<1.
Multiplying by x gives x/(1+x²)=Σ(−1)ⁿx²ⁿ⁺¹ on the same open interval. Multiplication changes powers, not the fact that the original geometric argument was −x².
At x=±1 the terms of these series fail to tend to zero, so both endpoints are excluded. The rational functions themselves exist there, but these particular series do not converge there.
A worked example, step by step
Find a power series for 1/(1−2x), including its interval.
- Use the geometric identity with u=2x.
- The series is Σₙ₌₀∞2ⁿxⁿ=1+2x+4x²+… .
- Require abs(2x)<1, giving abs(x)<1/2.
- At ±1/2 the terms do not tend to zero, so the interval is (−1/2,1/2).
A function’s domain and the convergence domain of one of its series are not necessarily the same.
Why not use the series for 1/(1+x²) at x=2?
Compare with an explanation
Its geometric ratio is −4, outside the convergence condition, although the rational function itself is defined.
Predict. Change one thing. Explain.
Select the substitution case and vary x within the displayed interval. Compare the original geometric pattern with its new even powers and alternating coefficients.
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 questionFind a power series for x²/(1−x) and state its interval.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: Start with 1/(1−x)=Σₙ₌₀∞xⁿ.
- 1 point: Multiply by x² to get Σₙ₌₀∞xⁿ⁺².
- 1 point: The open condition remains abs(x)<1.
- 1 point: At x=±1 terms do not approach zero, so the interval is (−1,1).
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 must change along with the formula in substitution?
The convergence condition.
RECALL 2Can a function exist where its series diverges?
Yes.
RECALL 3How do you multiply a series by x²?
Increase each exponent by two.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How can substitution create a new power series?
- Substitute into both the formula and its convergence condition.
- 1/(1+x²)=Σ(−1)ⁿx²ⁿ for abs(x)<1.
- Check transformed endpoints separately.
Remember: A function’s domain and the convergence domain of one of its series are not necessarily the same.
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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