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, 4.7, 5.3, 10.1, 10.3) are on the Calculus BC roadmap.
Understanding
- I can recognise an alternating series, including one written with cos(nπ), and reject one written with (−1)²ⁿ. (Guide: “What an alternating series is”; Practice Q1, Q5)
- I can state both conditions of the alternating series test and say what the test concludes. (Guide: “The alternating series test”; Practice Q3)
- I can explain, using partial sums, why decreasing terms that tend to 0 make an alternating series converge. (Guide: “Why it works”, Figure 1)
- I can explain why the test never proves divergence. (Guide: Worked example 2; Practice Q3, Q6(b))
Skills
- I can show that aₙ decreases by comparing aₙ₊₁ with aₙ directly. (Guide: “Showing that the terms decrease”; Practice Q5)
- I can show that aₙ decreases from some n on by using the sign of a derivative. (Guide: Worked example 1; Practice Q2, Q6(c), Q7(a))
- I can find lim aₙ, using L’Hospital’s Rule when it is needed. (Guide: Worked example 2(b); Practice Q6(c), Q7(b))
- I can find the values of a constant k for which the test applies, as in Σ (−1)ⁿ nᵏ/(n² + 1). (Practice Q4)
Reasoning
- I can switch to the nth term test when lim aₙ ≠ 0, and name it correctly. (Guide: Worked example 2(a); Practice Q6(a))
- I can write a full justification: positive aₙ, decreasing with a reason, limit 0, and a conclusion naming the test. (Guide: “Writing a full justification”; Practice Q5, Q7(c))
- I can explain why terms that rise at first do not stop the test from applying. (Practice Q2, Q7(d))
- I can tell the difference between the alternating series converging and Σ aₙ converging. (Guide: Common misconceptions; Practice Q2(D), Q7(e))
All ticked? Move on to Topic 10.8, the ratio test study guide, or return to the Calculus BC roadmap.