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, 10.13 and 10.14) are on the Calculus BC roadmap.
Understanding
- I can explain why changing a known series gives the Maclaurin series of the new function, without computing its derivatives. (Guide: “The idea: change a series you already know”)
- I can explain which operations keep the radius of convergence and why the endpoints must still be tested. (Guide: “Term-by-term differentiation and integration”; Practice Q4)
- I can explain why a series may fail to represent a function at a point where the function itself exists. (Guide: Common misconceptions)
Skills
- I can rewrite a rational function such as 1/(1 + 4x) or 5/(3 + x) in the form a/(1 − r) and write its series and interval. (Guide: Worked example 1; Practice Q1, Q5(a))
- I can substitute an expression such as 2x, −x² or −t⁴ into a known series, including inside every power. (Guide: “Substituting and multiplying by powers of x”; Practice Q2, Q6(a), Q7(a))
- I can multiply or divide a series by a power of x and pick out a single coefficient. (Guide: “Substituting and multiplying by powers of x”; Practice Q2)
- I can differentiate a power series term by term and start the new sum at the right index. (Guide: “Term-by-term differentiation and integration”; Practice Q3, Q5(b))
- I can integrate a power series term by term and find the constant from the value at the centre. (Guide: Worked example 2; Practice Q6(b), Q7(b))
- I can find the interval of convergence of a new series, testing each endpoint with a named test. (Guide: Worked example 2(b); Practice Q4, Q7(c))
Reasoning
- I can find f⁽ⁿ⁾(0) from a coefficient of a constructed series. (Guide: “Reading a derivative from a built series”; Practice Q6(d), Q7(d))
- I can find the exact sum of a numerical series by recognising it as a power series at a point inside its interval. (Guide: “Example: differentiating to sum a series”; Practice Q5(c))
- I can use a few terms of a constructed series to approximate a value and justify an error bound. (Guide: Worked example 2(c); Practice Q6(c), Q7(e))
All ticked? This is the last topic of Calculus BC. Return to the Calculus BC roadmap to plan your review of the whole course.