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, 10.1 and 10.2) are on the Calculus BC roadmap.
Understanding
- I can explain the difference between the sequence of terms aₙ and the sequence of partial sums Sₙ, and say which one decides convergence. (Guide: “Two sequences, not one”; Practice Q7(d))
- I can show, using aₙ = Sₙ − Sₙ₋₁, why the terms of a convergent series must approach 0. (Guide: “Why the terms of a convergent series must go to 0”)
- I can state the nth term test with both of its outcomes: “diverges” or “no conclusion”. (Practice Q2)
- I can give an example of a series whose terms approach 0 but which still diverges. (Guide: Worked example 3; Practice Q2)
Skills
- I can find the limit of a rational term by comparing the highest powers. (Guide: Worked example 1; Practice Q1, Q5(a), Q5(c))
- I can recognise an oscillating term whose limit does not exist, and conclude divergence. (Guide: Worked example 1(b); Practice Q5(b))
- I can use logarithms or L’Hospital’s Rule to find limits such as (1 + 2/n)ⁿ and n sin(3/n). (Guide: Worked example 2; Practice Q3)
- I can use the fact that a convergent series has terms approaching 0 to find a limit, such as lim (6aₙ + 1)/(aₙ + 2). (Guide: “Using the fact forwards”; Practice Q4)
- I can find the value of a constant that makes the limit of the terms equal to 0, and say what the test then shows. (Practice Q6(b))
Reasoning
- I can write a complete divergence justification: the limit of the terms, a comparison with 0, the name of the test and the conclusion. (Guide: “How to use the test, step by step”; Practice Q5, Q6(a), Q6(d))
- I can explain why “the terms approach 0” never proves that a series converges. (Practice Q2, Q6(c))
- I can interpret the nth term test in a context, explaining why a total grows without bound when the amounts added level off at a non-zero value. (Practice Q7(a), Q7(d))
All ticked? Move on to Topic 10.4, Integral Test for Convergence, or return to the Calculus BC roadmap.