When does a repeated fraction have a finite sum?
You will be able to: Derive the geometric sum and check its ratio condition.
When does a repeated fraction have a finite sum?
A cube is filled with slabs occupying half its volume, then a quarter, then an eighth. Each new slab occupies half as much volume as the last.
A useful starting point: What does adding infinitely many terms mean? →
Words and symbols before equations
- Geometric series
- Each term equals the previous term multiplied by a fixed ratio r.
- First term a
- The first actual quantity being added.
- Common ratio r
- The multiplier from one term to the next.
- Absolute value abs(r)
- The magnitude of the multiplier, ignoring its sign.
What this picture assumes
A unit cube has width, height and depth 1 unit. Each slab has full height and depth, and width 2⁻ⁿ. Slab volume is 2⁻ⁿ cubic units. No negative volumes are represented; every finite N leaves a gap.
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.
- 3 slabs fill 0.875 cubic units. The remaining 0.125 cubic units is positive at every finite N. Its limit is zero. The strip and optional cube show the same partition.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
For Sₙ=a+ar+…+arⁿ⁻¹, subtract rSₙ from Sₙ. Middle terms cancel, leaving (1−r)Sₙ=a(1−rⁿ).
When abs(r)<1, rⁿ→0, so the infinite sum is a/(1−r). For nonzero a and abs(r)≥1, the terms do not approach zero and the series diverges.
The optional cube partitions a unit volume into successive positive slabs. After N slabs, volume 1−2⁻ᴺ is filled and 2⁻ᴺ remains. It represents r=1/2 only; negative ratios need signed sums rather than physical volume.
A worked example, step by step
Find 3+3/2+3/4+… and its first-four-term approximation.
- The first term is a=3 and ratio r=1/2.
- Since abs(r)<1, the series converges.
- Its sum is 3/(1−1/2)=6.
- S₄=3(1−(1/2)⁴)/(1−1/2)=45/8=5.625; the remaining sum is 3/8.
Use the first term actually included, and verify abs(r)<1 before using the infinite-sum formula.
Does a finite number of slabs fill the cube exactly?
Compare with an explanation
No. The gap is 2⁻ᴺ for every finite N, and tends to zero in the limit.
Predict. Change one thing. Explain.
Add slabs one at a time. Predict the remaining volume before moving N. Rotate the optional cube to inspect how each slab spans the full height and depth; compare with the 2D strip.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
3 slabs fill 0.875 cubic units. The remaining 0.125 cubic units is positive at every finite N. Its limit is zero. The strip and optional cube show the same partition.
A unit cube has width, height and depth 1 unit. Each slab has full height and depth, and width 2⁻ⁿ. Slab volume is 2⁻ⁿ cubic units. No negative volumes are represented; every finite N leaves a gap.
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 questionA unit cube receives slabs of volume 1/2ⁿ for n=1,2,… . Find the total after three slabs, the remaining volume, and the infinite total.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: S₃=1/2+1/4+1/8=7/8.
- 1 point: Remaining volume is 1/8.
- 1 point: The ratio is 1/2 with magnitude below 1.
- 1 point: The infinite sum is (1/2)/(1−1/2)=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 1Why does the geometric formula work?
Subtracting rSₙ cancels the middle terms.
RECALL 2What is the convergence condition?
abs(r)<1, except the trivial all-zero series.
RECALL 3Why is 3D useful here?
It makes each slab’s width, height, depth and volume visible.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
When does a repeated fraction have a finite sum?
- Sₙ=a(1−rⁿ)/(1−r), r≠1.
- S=a/(1−r) if abs(r)<1.
- For the unit cube, Sₙ=1−2⁻ⁿ and remainder=2⁻ⁿ.
Remember: Use the first term actually included, and verify abs(r)<1 before using the infinite-sum formula.
Conditions: A unit cube has width, height and depth 1 unit. Each slab has full height and depth, and width 2⁻ⁿ. Slab volume is 2⁻ⁿ cubic units. No negative volumes are represented; every finite N leaves a gap.
Refresh Kid · AP Calculus BC Unit 10 · Objectives LIM-7.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 10.2, LIM-7.A. 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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