BC-only material. Work through the list without notes. If you cannot do a statement, follow its link, then come back. Guide = study guide; Practice = practice questions. Prerequisites (Topics 1.15 and 10.1, and exponent and log laws) are on the Calculus BC roadmap.
Understanding
- I can decide whether a series is geometric by checking for a constant ratio between successive terms. (Guide: “What makes a series geometric”)
- I can derive Sₙ = a(1 − rⁿ)/(1 − r) by subtracting r·Sₙ from Sₙ. (Guide: “The partial sum formula”)
- I can explain, using lim rⁿ and the definition of convergence, why a geometric series converges when |r| < 1 and diverges when |r| ≥ 1. (Guide: “When does a geometric series converge?”; Practice Q4)
Skills
- I can find the first term and common ratio of a geometric series whatever its starting index, and find its sum. (Guide: Worked example 1; Practice Q1, Q2)
- I can rewrite terms such as 3ⁿ⁺¹/5ⁿ or 2²ⁿ/5ⁿ in the form (constant)·(ratio)ⁿ. (Guide: Worked example 1; Practice Q4)
- I can find the values of x for which a geometric series with a ratio containing x converges, excluding the endpoints. (Guide: Worked example 3; Practice Q3)
- I can write a repeating decimal as a geometric series and as a fraction. (Guide: “Repeating decimals are geometric series”; Practice Q5)
- I can use the partial-sum formula, and solve for the number of terms needed to reach a target. (Guide: Worked example 2; Practice Q6(b)–(c), Q7(b))
- I can find a missing ratio from a given first term and sum. (Practice Q6(a), Q7(d))
Reasoning
- I can state the condition |r| < 1 before using a/(1 − r), and explain why the formula fails otherwise. (Guide: Worked example 1(b); Practice Q6(d))
- I can tell “the first n terms” (Sₙ) from “the total” (the full sum) in a context. (Guide: Worked example 2; Practice Q7(a)–(b))
- I can justify a claim about a total by comparing it with the sum of a geometric series. (Practice Q7(c))
All ticked? Move on to Topic 10.3, The nth Term Test for Divergence, or return to the Calculus BC roadmap.