Which four Maclaurin series are essential building blocks?
You will be able to: Recognize geometric, exponential, sine and cosine expansions with their domains.
Which four Maclaurin series are essential building blocks?
A few familiar patterns can build many more complicated functions. Start with functions whose derivatives repeat or whose geometric sum is already known.
A useful starting point: Why must each endpoint be tested separately? →
Words and symbols before equations
- Maclaurin series
- A Taylor series centered at zero.
- Coefficient cₙ
- The constant multiplying xⁿ.
- Even powers
- Powers 0,2,4,… .
- Odd powers
- Powers 1,3,5,… .
What this picture assumes
N counts nonzero terms. Sine and cosine use radians. Their series and the exponential series converge for all real x; the geometric series represents its function only for abs(x)<1.
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.
- eˣ. At x=0.5, 3 nonzero terms give 1.625. Function value: 1.648721. The full series equals the function at this input.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
The geometric identity gives 1/(1−x)=Σₙ₌₀∞xⁿ for abs(x)<1. The restriction is essential.
Since every derivative of eˣ at 0 equals 1, eˣ=Σₙ₌₀∞xⁿ/n! for all real x. Derivatives of sine and cosine repeat in cycles, giving sin x=Σ(−1)ⁿx²ⁿ⁺¹/(2n+1)! and cos x=Σ(−1)ⁿx²ⁿ/(2n)!, also for all real x.
These are function equalities on the stated domains, justified by convergence and remainder arguments. Remember both the pattern of powers and the factorial indices; use radians for trigonometric derivatives.
A worked example, step by step
Write the first three nonzero terms of sin x and cos x.
- For sine, substitute n=0,1,2 into (−1)ⁿx²ⁿ⁺¹/(2n+1)!.
- The terms are x−x³/6+x⁵/120.
- For cosine, use (−1)ⁿx²ⁿ/(2n)!.
- The terms are 1−x²/2+x⁴/24; both full series converge for every real x.
A truncated expression is an approximation. An infinite equality also needs its domain.
Why does the sine series have no constant term?
Compare with an explanation
sin(0)=0, and its derivative cycle leaves only odd powers.
Predict. Change one thing. Explain.
Choose a standard function and compare its finite polynomial with the function. For the geometric case, compare inputs inside and outside abs(x)<1 and explain the domain restriction.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
eˣ. At x=0.5, 3 nonzero terms give 1.625. Function value: 1.648721. The full series equals the function at this input.
N counts nonzero terms. Sine and cosine use radians. Their series and the exponential series converge for all real x; the geometric series represents its function only for abs(x)<1.
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 questionWrite the first four terms of eˣ and the first three nonzero terms of sin x. State both convergence domains.
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Compare with the answer and four-point rubric
- 1 point: eˣ begins 1+x+x²/2+x³/6.
- 1 point: sin x begins x−x³/6+x⁵/120.
- 1 point: The full exponential series converges to eˣ for every real x.
- 1 point: The full sine series converges to sin x for every real x, with radians used.
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 geometric-series domain?
abs(x)<1 for Σxⁿ.
RECALL 2Which powers occur in cosine?
Even powers, including the constant term.
RECALL 3Which standard series here have infinite radius?
Exponential, sine and cosine.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Which four Maclaurin series are essential building blocks?
- 1/(1−x)=Σxⁿ, abs(x)<1.
- eˣ=Σxⁿ/n!, all real x.
- sin x=Σ(−1)ⁿx²ⁿ⁺¹/(2n+1)!; cos x=Σ(−1)ⁿx²ⁿ/(2n)!, all real x.
Remember: A truncated expression is an approximation. An infinite equality also needs its domain.
Conditions: N counts nonzero terms. Sine and cosine use radians. Their series and the exponential series converge for all real x; the geometric series represents its function only for abs(x)<1.
Refresh Kid · AP Calculus BC Unit 10 · Objectives LIM-8.E, LIM-8.F · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 10.14, LIM-8.E, LIM-8.F. 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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