AP CALCULUS BC Scoring Guide
10.9
1. Which of the following series are conditionally convergent?
(A) I only
(B) I and II only
(C) I and III only
(D) II and III only
Answer C
This option is correct.
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Scoring Guide
10.9
2. Which of the following series is conditionally convergent?
(A)
(B)
(C)
(D)
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Scoring Guide
10.9
Answer C
Correct. The series converges by the alternating series test:
; if , then for , showing that the
terms decrease as increases.
The series diverges, however, by the comparison test with the divergent -series
, since for all . Therefore, the series is
conditionally convergent.
3.
For what values of is the series conditionally convergent?
(A)
(B)
(C) only
(D) only
Answer C
Correct. For , or , the series is an alternating series with individual
terms that decrease in absolute value to 0. Therefore, converges for by the
alternating series test.
The series is a -series and therefore diverges for , or . Since
diverges for , the series diverges for by the limit
comparison test.
Since the series converges for and the series of absolute values
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Scoring Guide
10.9
diverges for , is conditionally convergent for .
4.
Which of the following statements is true about the series ?
(A) The series converges conditionally.
(B) The series converges absolutely.
(C) The series converges but neither conditionally nor absolutely.
(D) The series diverges.
Answer A
Correct. The series is an alternating series with individual terms that decreases in absolute
value to 0. Therefore, it converges by the alternating series test. The series of absolute values
diverges, as it is a -series with .
Therefore, is conditionally convergent.
5.
Consider the series and . Which of the following statements is true?
(A) Both series converge absolutely.
(B) Both series converge conditionally.
(C) converges absolutely, and converges conditionally.
(D) converges conditionally, and converges absolutely.
Answer D
Correct. A series is conditionally convergent if the series converges but the series of absolute
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Scoring Guide
10.9
terms diverges. It is absolutely convergent if the series converges and the series of absolute
terms also converges.
Each of the series in this problem converges by the alternating series test.
The series converges conditionally because the series diverges, since it is a
-series with .
The series converges absolutely because the series converges, since it is a -series
with .
6.
Consider the series . Which of the following statements is true?
(A) The series converges absolutely.
(B) The series converges conditionally.
(C) The series diverges.
(D) It cannot be determined whether the series converges or diverges from the information given.
Answer C
Correct. The series diverges by the term test. As , approaches .
Therefore, as , the terms are alternating between values approaching and . This
shows that does not exist. Therefore, since the terms do not approach in the limit as
, the series does not converge.
7. Which of the following series is conditionally convergent?
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Scoring Guide
10.9
(A)
(B)
(C)
(D)
8. Which of the following series are conditionally convergent?
I.
II.
III.
(A) I only
(B) II only
(C) II and III only
(D) I, II, and III
Answer C
Correct. A series is conditionally convergent if the series converges but the series of absolute
terms diverges. Each of the three series in this problem converges by the alternating series test.
The series is not conditionally convergent, since converges by the ratio test (so this
series is absolutely convergent).
The series is conditionally convergent because the series diverges, since it is a
-series with .
The series is conditionally convergent because the series diverges by the limit
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10.9
comparison test with the harmonic series .
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