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 and 6.12, and the laws of logarithms) are on the Calculus BC roadmap.
Understanding
- I can explain the difference between the terms aₙ, the series Σ aₙ and the partial sums Sₙ. (Guide: “Three objects you must keep apart”; Practice Q1)
- I can state, in my own words, what it means for a series to converge to S, using the limit of the partial sums. (Guide: “The definition: convergence is a limit”; Practice Q2)
- I can describe the two ways a series can diverge: partial sums that grow without bound, and partial sums that never settle. (Guide: “The definition: convergence is a limit”; Practice Q4)
Skills
- I can write out and evaluate a partial sum such as S₃, counting terms from the correct starting index. (Guide: “Three objects you must keep apart”; Practice Q1, Q7(a))
- I can find a₁ = S₁ and aₙ = Sₙ − Sₙ₋₁ from a formula for the partial sums. (Guide: Worked example 1; Practice Q6(a))
- I can find the sum of a series from a formula for Sₙ by taking a limit. (Guide: Worked example 1; Practice Q2, Q6(b))
- I can split a term with partial fractions or log laws, find a telescoping Sₙ and take its limit. (Guide: Worked examples 2 and 3; Practice Q3, Q5, Q7(b)–(c))
- I can find the sum of a series that starts at a later index by subtracting the missing terms. (Guide: “Facts that follow from the definition”; Practice Q6(c))
- I can find how many terms are needed for Sₙ to be within a given distance of the sum. (Practice Q7(d))
Reasoning
- I can justify a conclusion about convergence by referring to lim Sₙ, not to lim aₙ. (Practice Q2, Q5(c), Q6(b))
- I can give a counter-example to show that terms tending to 0 do not guarantee convergence. (Guide: Worked example 3; Practice Q6(d))
- I can explain why a table of partial sums is evidence, not proof. (Guide: “Seeing convergence on a graph”; Common misconceptions)
All ticked? Move on to Topic 10.2, Working with Geometric Series, or return to the Calculus BC roadmap.