What can the limit of a term tell us?
You will be able to: Use the nth-term test without reversing its logic.
What can the limit of a term tell us?
If you keep adding about one dollar each week, your total cannot settle at a finite amount. Additions must shrink toward zero for a finite infinite total to be possible.
A useful starting point: How do starting indices and negative ratios change a sum? →
Words and symbols before equations
- Necessary condition
- A condition that must hold, but may not be enough.
- Divergence test
- If aₙ does not tend to zero, Σaₙ diverges.
- Inconclusive
- This test alone gives no decision.
- Limit does not exist
- Terms fail to approach one number.
What this picture assumes
Terms start at n=1. The nth-term test rules out convergence for a nonzero or nonexistent term limit. A zero limit is inconclusive.
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.
- At N=6, a_N=0.166667, S_N=2.45. Term limit 0: the nth-term test is inconclusive. Points are discrete; finite samples are not a proof.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
If Sₙ→S, then aₙ=Sₙ−Sₙ₋₁→S−S=0. This explains why convergent series must have terms approaching zero.
For aₙ=n/(n+1), divide by n to get 1/(1+1/n)→1. Since this limit is not zero, the series diverges.
If aₙ→0, stop short of concluding convergence. Both Σ1/n² and Σ1/n have terms tending to zero, but only the first converges. Oscillatory terms such as (−1)ⁿ also fail the necessary condition.
A worked example, step by step
Decide what the nth-term test says about Σ(3n+1)/(2n+5).
- Examine the term aₙ, not a partial sum.
- Divide numerator and denominator by n.
- The limit is (3+1/n)/(2+5/n)→3/2.
- Because 3/2≠0, the series diverges.
The statement “term limit zero, therefore converges” is invalid.
What does the test say about Σ1/n?
Compare with an explanation
It is inconclusive because 1/n→0; another argument proves divergence.
Predict. Change one thing. Explain.
Compare the three term sequences. Identify which ones fail the necessary condition and which require another test. Do not confuse a small displayed term with a proven zero limit.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
At N=6, a_N=0.166667, S_N=2.45. Term limit 0: the nth-term test is inconclusive. Points are discrete; finite samples are not a proof.
Terms start at n=1. The nth-term test rules out convergence for a nonzero or nonexistent term limit. A zero limit is inconclusive.
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 questionApply only the nth-term test to Σn²/(n²+1) and Σ1/n². Explain the different conclusions.
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Compare with the answer and four-point rubric
- 1 point: n²/(n²+1)→1.
- 1 point: The first series diverges because the term limit is nonzero.
- 1 point: 1/n²→0.
- 1 point: The nth-term test is inconclusive for the second; a p-series test is needed.
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 must aₙ→0 for convergence?
aₙ=Sₙ−Sₙ₋₁ and both partial sums approach the same limit.
RECALL 2What does zero term limit prove?
Only that the divergence test does not rule out convergence.
RECALL 3What if the term limit does not exist?
The series diverges.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
What can the limit of a term tell us?
- If lim aₙ≠0 or does not exist, Σaₙ diverges.
- If lim aₙ=0, this test is inconclusive.
- Convergent series ⇒ terms tend to zero.
Remember: The statement “term limit zero, therefore converges” is invalid.
Conditions: Terms start at n=1. The nth-term test rules out convergence for a nonzero or nonexistent term limit. A zero limit is inconclusive.
Refresh Kid · AP Calculus BC Unit 10 · Objectives LIM-7.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 10.3, 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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