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 10.2, 10.5, 10.7, 10.8 and 10.9) are on the Calculus BC roadmap.
Understanding
- I can write the general form of a power series and name its centre and coefficients. (Guide: “What a power series is”)
- I can explain the three possible outcomes: convergence only at the centre, on an interval centred at c, or for every real x. (Guide: “Only three things can happen”; Practice Q3)
- I can explain why the ratio test cannot decide an endpoint. (Guide: “Endpoint toolkit”)
- I can explain why a power series with positive radius is the Taylor series of its sum. (Guide: “Power series and Taylor series”)
Skills
- I can use the ratio test on the whole term to find a radius of convergence, including R = 0 and R = ∞. (Guide: “Finding the radius with the ratio test”; Practice Q1, Q3)
- I can rewrite a condition such as |2x − 5| < 1 in the form |x − c| < R before reading off R. (Guide: Worked example 2)
- I can handle series with only even powers, such as Σ x^(2n)/(4ⁿ(n + 1)). (Practice Q5)
- I can test each endpoint with a named test (p-series, alternating series, nth term, comparison) and write the interval with correct brackets. (Guide: Worked example 1; Practice Q2, Q5, Q6(b))
- I can find f⁽ⁿ⁾(c) from a coefficient using f⁽ⁿ⁾(c) = n! · aₙ. (Practice Q4)
Reasoning
- I can state that term-by-term differentiation or integration keeps the radius, and retest the endpoints of the new series. (Guide: “Differentiating and integrating term by term”; Practice Q6(c)–(d))
- I can decide where a series must converge, must diverge, or might do either, from convergence at one point and divergence at another. (Guide: “Only three things can happen”; Practice Q7)
- I can justify each endpoint decision in a sentence that names the test and its conditions. (Practice Q6, Q7(c))
All ticked? Move on to Topic 10.14, Finding Taylor or Maclaurin series for a function, or return to the Calculus BC roadmap.