How do starting indices and negative ratios change a sum?
You will be able to: Identify the first term and ratio in indexed geometric series.
How do starting indices and negative ratios change a sum?
A savings plan makes its first payment this month. A formula beginning at month zero and one beginning at month one can describe different first payments.
A useful starting point: When does a repeated fraction have a finite sum? →
Words and symbols before equations
- Index
- The integer labeling each term.
- Lower index
- The first integer included in the sum.
- Alternation
- Successive terms switch sign.
- Finite prefix
- A finite number of beginning terms.
What this picture assumes
First term a=1 at index n=0. N terms means exponents 0 through N−1. Values are dimensionless. Only abs(r)<1 gives a finite infinite sum.
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.
- r=0.5, N=6, last included term=0.03125, S_N=1.96875. abs(r)<1: infinite sum 1/(1−r)=2.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
In Σₙ₌₂∞3(1/2)ⁿ, substitute n=2 first: a=3/4. The multiplier between successive terms is still 1/2.
In Σₙ₌₀∞(−1/2)ⁿ, a=1 and r=−1/2. Partial sums alternate around 2/3; a negative ratio does not prevent convergence when its magnitude is below 1.
Adding or removing finitely many terms changes a convergent series sum, but not whether its tail converges. Always state the starting index when reporting a value.
A worked example, step by step
Evaluate Σₙ₌₃∞2(−1/3)ⁿ.
- At n=3, the first term is 2(−1/3)³=−2/27.
- The ratio is −1/3.
- Its magnitude is below 1, so use the geometric formula.
- S=(−2/27)/(1+1/3)=−1/18.
The coefficient in front of rⁿ is not always the first term.
What is the first term of Σₙ₌₂∞5(1/3)ⁿ?
Compare with an explanation
5/9, found by substituting the lower index 2.
Predict. Change one thing. Explain.
Move r between negative and positive values. Observe alternating partial sums for negative r and inspect r=±1. Explain why the finite display must be paired with the ratio condition.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
r=0.5, N=6, last included term=0.03125, S_N=1.96875. abs(r)<1: infinite sum 1/(1−r)=2.
First term a=1 at index n=0. N terms means exponents 0 through N−1. Values are dimensionless. Only abs(r)<1 gives a finite infinite sum.
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 questionFind Σₙ₌₂∞4(1/3)ⁿ, and explain how it differs from the sum starting at n=0.
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Compare with the answer and four-point rubric
- 1 point: The first included term is 4/9.
- 1 point: The ratio is 1/3, so convergence holds.
- 1 point: The requested sum is (4/9)/(2/3)=2/3.
- 1 point: Starting at 0 gives 6; subtracting the omitted terms 4 and 4/3 leaves 2/3.
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 1How do you find a first term?
Substitute the lower index.
RECALL 2Does a negative ratio always diverge?
No; its magnitude decides geometric convergence.
RECALL 3Can deleting ten terms fix a divergent tail?
No.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How do starting indices and negative ratios change a sum?
- Σₙ₌ₘ∞crⁿ=crᵐ/(1−r), abs(r)<1.
- Negative r gives alternating signs.
- Changing finitely many terms does not change convergence.
Remember: The coefficient in front of rⁿ is not always the first term.
Conditions: First term a=1 at index n=0. N terms means exponents 0 through N−1. Values are dimensionless. Only abs(r)<1 gives a finite infinite sum.
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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