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 5.3, 6.9, 6.11, 6.13 and 10.3) are on the Calculus BC roadmap.
Understanding
- I can state the integral test, with its three conditions and its “both converge or both diverge” conclusion. (Guide: “The integral test”; Practice Q1)
- I can explain, using rectangles above and below the curve, why a finite integral means a convergent series and an infinite integral means a divergent one. (Guide: Figure 1 and “Why the two pictures settle the question”)
- I can explain why the value of the integral is not the sum of the series. (Guide: Worked example 1; Practice Q2, Q7(d))
- I can recognise series where the test does not apply: terms that change sign or terms that are not decreasing. (Guide: “When the integral test does not fit”; Practice Q3)
Skills
- I can show that f is decreasing using f′(x) < 0, and find the point N from which it decreases. (Guide: Worked example 2; Practice Q6(b), Q7(a))
- I can evaluate an improper integral by substitution and take the limit as b → ∞. (Guide: Worked examples 1 and 2; Practice Q6(c))
- I can evaluate an improper integral by parts or with arctan, including limits such as b e^(−b) → 0. (Guide: Worked example 3; Practice Q5, Q7(b))
- I can find the values of a constant for which a series converges, by treating the cases of the antiderivative separately. (Practice Q4)
Reasoning
- I can write a complete integral-test justification: the conditions with reasons, the improper integral as a limit, and the conclusion naming the test. (Guide: Worked examples 1–3; Practice Q5, Q6)
- I can use the nth term test first and explain why it is inconclusive before moving to the integral test. (Practice Q6(a))
- I can use partial sums to show that a claimed sum equal to the integral is wrong. (Practice Q7(d))
All ticked? Move on to Topic 10.5, Harmonic Series and p-Series, or return to the Calculus BC roadmap.