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, 10.7 and 10.9) are on the Calculus BC roadmap.
Understanding
- I can state the alternating series error bound, |S − Sₙ| ≤ aₙ₊₁, and the three conditions it needs. (Guide: “The alternating series error bound”; Practice Q1)
- I can explain, with the zigzag picture, why the sum lies between two consecutive partial sums. (Guide: Figure 1; Practice Q4)
- I can explain the difference between the error bound (a guaranteed maximum) and the actual error. (Guide: Worked example 1; Common misconceptions)
Skills
- I can check the conditions of the alternating series test before using the bound. (Guide: Worked example 1(a); Practice Q5(a))
- I can find the bound for a given partial sum, using the first omitted term. (Practice Q1, Q5(c))
- I can show that an approximation is within a given tolerance by computing the bound and comparing it with the tolerance. (Guide: Worked example 1(c); Practice Q5(c))
- I can find the least number of terms that guarantees a required accuracy, and check the values either side. (Guide: Worked example 2; Practice Q2, Q6(a), Q7(d))
- I can compute a partial sum exactly or to 6 decimal places without rounding too early. (Practice Q5(b), Q6(b))
Reasoning
- I can decide whether Sₙ is an overestimate or an underestimate from the sign of the first omitted term. (Guide: Worked example 1(d); Practice Q3, Q5(d))
- I can give an interval that must contain the sum. (Practice Q4, Q5(d), Q6(c))
- I can explain when the bound may be used for a series whose terms decrease only after a certain index. (Guide: “When the bound does not apply”; Practice Q7(c))
All ticked? Move on to Topic 10.11, Finding Taylor Polynomial Approximations of Functions, or return to the Calculus BC roadmap.