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 and 10.5) are on the Calculus BC roadmap.
Understanding
- I can state the direct comparison test, including the condition that the terms are non-negative from some n onwards. (Guide: “The direct comparison test”; Practice Q1)
- I can name the two directions of direct comparison that give a conclusion and the two that give none. (Guide: Figure 1; Practice Q1, Q3)
- I can state the limit comparison test and explain why it needs 0 < L < ∞. (Guide: “The limit comparison test”; Practice Q2)
- I can explain why positive terms matter for both tests. (Guide: “The idea: compare with a series you know”)
Skills
- I can choose a p-series or geometric benchmark by keeping the dominant terms. (Guide: “Choosing bₙ”; Practice Q4)
- I can build a direct comparison using bounded pieces such as sin n or arctan n. (Guide: Worked example 1; Practice Q6(a))
- I can show a series diverges by proving its terms are bigger than those of a divergent series. (Guide: Worked example 1(b); Practice Q6(b))
- I can find lim aₙ/bₙ by dividing by the highest power, including with roots and exponentials. (Guide: Worked examples 2 and 3; Practice Q2, Q5(a), Q7(b))
- I can use a geometric comparison to give an upper bound for the sum of a series. (Guide: Worked example 3(a); Practice Q7(c))
Reasoning
- I can recognise when a natural inequality points the wrong way and switch to limit comparison or a better inequality. (Guide: Worked example 2(a); Practice Q5(b), (c), Q7(a))
- I can explain why a “smaller than divergent” comparison gives no conclusion. (Practice Q3, Q6(c))
- I can write a complete justification: positivity, benchmark with its reason, inequality or limit, name of the test, conclusion. (Guide: “Writing a complete justification”; Practice Q5–Q7)
All ticked? Move on to Topic 10.7, Alternating Series Test for Convergence, or return to the Calculus BC roadmap.