Free response without a calculator (Part B)
BC: Choosing convergence tests
- Units 6 and 10
- 9 points
- About 15 minutes
- BC only
A multi-part problem you solve by hand, often from a graph, a table, an equation or a differential equation. You show your work, use exact values, and justify answers with calculus reasons such as a sign change in a derivative or the conditions of a theorem. On the exam: 4 questions in Part B of the free-response section (60 minutes, no calculator). The whole free-response section is 6 questions in 90 minutes and counts for 50% of the score. AB and BC use the same format; on BC, Part B usually includes a series question and a question on BC-only topics.
The question
Answer each part without a calculator. For each series, name any test you use and show that its conditions are met.
Suggested time: 15 minutes
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Part (a)
2 pointsDetermine whether the series Σ (n = 1 to ∞) 4(−1/3)ⁿ⁻¹ converges or diverges. If it converges, find its sum.
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Part (b)
3 pointsUse the integral test to determine whether the series Σ (n = 2 to ∞) 1/(n(ln n)²) converges or diverges. Use limit notation for the improper integral.
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Part (c)
3 pointsDetermine whether the series Σ (n = 1 to ∞) (−1)ⁿ·n/(n² + 1) converges absolutely, converges conditionally, or diverges. Justify your answer.
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Part (d)
1 pointDoes the series Σ (n = 1 to ∞) (3n + 1)/(2n + 5) converge or diverge? Give a reason for your answer.
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