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 10.3 to 10.8) are on the Calculus BC roadmap.
Understanding
- I can define absolutely convergent, conditionally convergent and divergent using Σ aₙ and Σ |aₙ|. (Guide: “Two series hiding inside one”; Practice Q1)
- I can explain why a series that converges absolutely must converge, using 0 ≤ aₙ + |aₙ| ≤ 2|aₙ|. (Guide: “Why absolute convergence implies convergence”; Practice Q1)
- I can explain why the converse is false, with a counter-example. (Practice Q1, Q7(d))
- I can explain why only an absolutely convergent series can be regrouped or reordered without changing its sum. (Guide: “Regrouping and reordering terms”; Practice Q4, Q6(c))
Skills
- I can classify Σ (−1)ⁿ⁺¹/nᵖ for any value of p. (Guide: “A reference family”; Practice Q2)
- I can show that Σ |aₙ| diverges using limit comparison or the integral test. (Guide: Worked example 1; Practice Q5(a), Q7(b))
- I can show that Σ |aₙ| converges using comparison or limit comparison with a p-series. (Guide: Worked example 2; Practice Q6(b))
- I can use the ratio test on |aₙ| and say what L < 1 and L > 1 mean for Σ aₙ. (Guide: Worked example 3; Practice Q3)
- I can apply the alternating series test, showing that the terms decrease and approach 0. (Guide: Worked example 1; Practice Q5(b), Q7(c))
Reasoning
- I can follow the decision path (nth term test, then Σ |aₙ|, then Σ aₙ) and give a full written classification. (Guide: Figure 1; Practice Q3, Q5, Q7)
- I can handle a series whose signs do not alternate by testing its absolute values. (Guide: Worked example 2)
- I can recognise disguised alternating factors such as cos(πn) = (−1)ⁿ. (Practice Q6(a))
All ticked? Move on to Topic 10.10, Alternating Series Error Bound, or return to the Calculus BC roadmap.