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, 10.2, 10.3, 10.5, 10.6) are on the Calculus BC roadmap.
Understanding
- I can state the ratio test, including all three cases L < 1, L > 1 and L = 1. (Guide: “The ratio test”; Practice Q1)
- I can explain the link between the ratio test and geometric series. (Guide: “The idea” and “Why each case holds”)
- I can explain why L = 1 gives no conclusion, using Σ 1/n and Σ 1/n² as examples. (Guide: “When the ratio test fails”; Practice Q7(d))
- I can explain why only the limit of the ratios matters, not the first few ratios. (Guide: Worked example 1, Figure 1)
Skills
- I can form aₙ₊₁/aₙ the right way up and simplify powers such as 7ⁿ⁺¹/7ⁿ. (Guide: “Algebra you will need”; Practice Q1, Q6(a))
- I can cancel factorials, including (n + 1)!/n! and (2n + 2)!/(2n)!. (Guide: Worked example 2; Practice Q3)
- I can keep absolute value bars when the terms change sign. (Guide: Common misconceptions; Practice Q5)
- I can find the values of a constant for which the ratio test shows convergence. (Practice Q4)
- I can use the ratio test when the terms are given by a recursive rule. (Practice Q7(b), (c))
Reasoning
- I can write a conclusion that states L, compares it with 1 and names the ratio test. (Guide: Worked examples 1 and 2; Practice Q5)
- I can recognise series where the ratio test will give L = 1 and choose a better test. (Guide: “Choosing the ratio test”; Practice Q2, Q6(b), (c))
- I can explain why L > 1 means the terms cannot tend to 0. (Guide: “Why each case holds”; Practice Q7(c))
All ticked? Move on to Topic 10.9, the absolute and conditional convergence study guide, or return to the Calculus BC roadmap.