How many terms give a guaranteed alternating-series accuracy?
You will be able to: Bound a remainder by the next omitted magnitude and choose N.
How many terms give a guaranteed alternating-series accuracy?
If a sequence of corrections keeps overshooting by smaller amounts, stopping after one correction leaves an error no larger than the next correction.
A useful starting point: What is the difference between absolute and conditional convergence? →
Words and symbols before equations
- Remainder R_N
- The exact sum S minus its partial sum S_N.
- Error tolerance ε
- A specified positive upper limit for the absolute error.
- Next omitted term
- The first term not included in the approximation.
- Error bound
- A guarantee, not necessarily the actual error.
What this picture assumes
For these decreasing alternating series, S lies between S_N and S_(N+1) and abs(error)≤1/(N+1)ᵖ. The bound is not the actual error.
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.
- N=6, p=1: S_N=0.616667. The true sum lies in [0.616667, 0.759524]; error ≤ 0.142857. Conditional convergence: absolute series is harmonic. The bracket is guaranteed by decreasing magnitudes and a zero limit.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
When alternating magnitudes decrease to zero, abs(R_N)≤b_(N+1). The sum lies between S_N and S_(N+1). State these hypotheses before applying the bound.
For Σ(−1)ⁿ⁺¹/n² starting at n=1, after N terms the error is at most 1/(N+1)². Solve the inequality using N+1, not N.
A strict requested accuracy abs(error)<ε can be guaranteed by making the bound strictly less than ε. The exact error can be smaller, so this method gives a sufficient term count, not necessarily the smallest possible count.
A worked example, step by step
How many terms guarantee error less than 0.01 for Σ(−1)ⁿ⁺¹/n²?
- The magnitudes 1/n² decrease to zero.
- Require 1/(N+1)²<0.01.
- This gives N+1>10, so the smallest integer meeting this bound is N=10.
- With ten terms the bound is 1/121≈0.008265<0.01.
N included terms means use term N+1 for the bound; check whether the requested inequality is strict.
For an alternating series, why can you bracket S with consecutive sums?
Compare with an explanation
The decreasing corrections alternate sides of the limiting sum.
Predict. Change one thing. Explain.
Increase N and compare the two consecutive partial sums forming the bracket. Switch p between 1 and 2 and explain the different rates of narrowing.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
N=6, p=1: S_N=0.616667. The true sum lies in [0.616667, 0.759524]; error ≤ 0.142857. Conditional convergence: absolute series is harmonic. The bracket is guaranteed by decreasing magnitudes and a zero limit.
For these decreasing alternating series, S lies between S_N and S_(N+1) and abs(error)≤1/(N+1)ᵖ. The bound is not the actual error.
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 questionApproximate Σ(−1)ⁿ⁺¹/n² with S₂. Give its next-term bound and an interval containing the true sum.
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Compare with the answer and four-point rubric
- 1 point: S₂=1−1/4=3/4.
- 1 point: The next magnitude is 1/9, so abs(error)≤1/9.
- 1 point: S₃=3/4+1/9=31/36.
- 1 point: The sum lies in [3/4,31/36].
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 1Which term bounds the error?
The next omitted magnitude.
RECALL 2What conditions are required?
Alternating terms with decreasing magnitudes tending to zero.
RECALL 3Does the sufficient term count prove the smallest possible actual-error count?
No.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How many terms give a guaranteed alternating-series accuracy?
- abs(S−S_N)≤b_(N+1) under alternating-test conditions.
- S lies between consecutive partial sums.
- A bound is not the exact error.
Remember: N included terms means use term N+1 for the bound; check whether the requested inequality is strict.
Conditions: For these decreasing alternating series, S lies between S_N and S_(N+1) and abs(error)≤1/(N+1)ᵖ. The bound is not the actual error.
Refresh Kid · AP Calculus BC Unit 10 · Objectives LIM-7.B · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 10.10, LIM-7.B. 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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