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 10.1 to 10.4) are on the Calculus BC roadmap.
Understanding
- I can name the harmonic series, the alternating harmonic series and a p-series from their terms, and state whether each converges. (Guide: “Three named series”; Practice Q3, Q4)
- I can explain why the harmonic series diverges even though its terms approach 0, and why the nth term test says nothing about it. (Guide: “Why the harmonic series diverges”; Practice Q3)
- I can explain how the alternating harmonic series can converge when the harmonic series does not. (Guide: “The alternating harmonic series”; Practice Q6(d))
- I can tell a p-series (variable in the base) from a geometric series (variable in the power). (Guide: “Recognising p-series in disguise”)
Skills
- I can rewrite a term such as n/(n² √n) as a single power 1/nᵖ and decide convergence from p. (Guide: Worked example 1; Practice Q1, Q5)
- I can find the values of a parameter that make a p-series converge. (Practice Q2)
- I can use constant multiples, shifts of the starting index and sums of series to classify series built from p-series. (Guide: Worked example 3; Practice Q4, Q5(d))
- I can compute partial sums of the harmonic and alternating harmonic series exactly. (Practice Q6(a))
Reasoning
- I can prove the p-series rule with the integral test, stating the conditions the test needs. (Guide: “The p-series rule and why it is true”; Practice Q7(a))
- I can use the grouping argument to show S(2ᵏ) ≥ 1 + k/2 and use it to guarantee a partial sum passes a target. (Guide: Worked example 2; Practice Q6(b), (c))
- I can bound the partial sums of a convergent p-series using rectangles under the curve. (Practice Q7(b))
- I can spot a wrong-direction argument, such as “bigger than a convergent series, so it converges”. (Practice Q7(c))
- I can write a full justification: name the series, state p and compare it with 1. (Guide: Worked example 1, “What to write”)
All ticked? Move on to Topic 10.6, Comparison Tests for Convergence, or return to the Calculus BC roadmap.