How can an area decide whether a series converges?
You will be able to: Check the integral-test conditions and connect rectangles to an improper integral.
How can an area decide whether a series converges?
Imagine stacking rectangles of width one under a decreasing positive curve. Their heights are the series terms; the nearby smooth area controls whether their total can stay finite.
A useful starting point: What can the limit of a term tell us? →
Words and symbols before equations
- Improper integral
- An integral over an unbounded interval, evaluated by a limit.
- Eventually decreasing
- Decreasing after some finite starting point.
- Positive function
- f(x)>0 on the tail used by the test.
- Tail Rₙ
- The sum left after the first N terms of a convergent series.
What this picture assumes
f(x)=1/x² is positive, continuous and decreasing for x≥1. Rectangle widths are 1. Right-endpoint rectangles on [n,n+1] lie below the curve. The exact series sum is not the integral value.
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.
- S_N=1.423611 for the first 4 series terms. The unseen tail is between 1/(N+1)=0.2 and 1/N=0.25. The rectangles shown on [1,N+1] use heights a₂ through a_(N+1). The integral from 1 to infinity is 1, not the series sum.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
Suppose aₙ=f(n), where f is positive, continuous and decreasing for x≥M. Then Σaₙ and ∫ₘ∞f(x)dx either both converge or both diverge.
For f(x)=1/x², an antiderivative is −1/x. The integral from 1 to b is 1−1/b→1, so Σ1/n² converges. This does not mean its sum equals 1.
For a positive decreasing f, rectangles also give Integral from N+1 to ∞ of f(x) dx ≤ R_N ≤ integral from N to ∞ of f(x) dx. The model shows these tail bounds for 1/x². Finite early terms do not affect the conclusion.
A worked example, step by step
Determine whether Σₙ₌₁∞1/n³ converges using an integral.
- Use f(x)=x⁻³; it is positive and continuous for x≥1.
- f′(x)=−3x⁻⁴<0, so it decreases.
- ∫₁ᵇx⁻³dx=1/2−1/(2b²).
- The limit is 1/2, finite, so the series converges; 1/2 is the integral value, not the series sum.
Do not apply the integral test to a sign-changing function, and do not equate a series sum with its comparison integral.
Must the integral equal the series sum?
Compare with an explanation
No. The test transfers convergence, not numerical equality.
Predict. Change one thing. Explain.
Increase N for 1/n². Compare the last rectangle, partial sum and integral tail bounds. Explain why both bounds shrink even though the partial sum increases.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
S_N=1.423611 for the first 4 series terms. The unseen tail is between 1/(N+1)=0.2 and 1/N=0.25. The rectangles shown on [1,N+1] use heights a₂ through a_(N+1). The integral from 1 to infinity is 1, not the series sum.
f(x)=1/x² is positive, continuous and decreasing for x≥1. Rectangle widths are 1. Right-endpoint rectangles on [n,n+1] lie below the curve. The exact series sum is not the integral value.
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 questionUse the integral test for Σₙ₌₁∞1/√n. Include the function conditions and the limiting integral.
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Compare with the answer and four-point rubric
- 1 point: f(x)=x⁻¹ᐟ² is positive and continuous for x≥1.
- 1 point: f′(x)=−(1/2)x⁻³ᐟ²<0.
- 1 point: ∫₁ᵇx⁻¹ᐟ²dx=2√b−2→∞.
- 1 point: Therefore the series 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 1Can the conditions start after n=1?
Yes; a finite prefix does not affect convergence.
RECALL 2Why compare rectangles with an integral?
Upper and lower area estimates constrain their total.
RECALL 3Does a convergent integral give the exact series sum?
Usually no.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How can an area decide whether a series converges?
- Positive, continuous, decreasing f with aₙ=f(n): series and integral share convergence.
- Evaluate the improper integral using a limit.
- For decreasing positive terms, Integral from N+1 to ∞ of f(x) dx ≤ R_N ≤ integral from N to ∞ of f(x) dx.
Remember: Do not apply the integral test to a sign-changing function, and do not equate a series sum with its comparison integral.
Conditions: f(x)=1/x² is positive, continuous and decreasing for x≥1. Rectangle widths are 1. Right-endpoint rectangles on [n,n+1] lie below the curve. The exact series sum is not the integral value.
Refresh Kid · AP Calculus BC Unit 10 · Objectives LIM-7.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 10.4, 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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