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AP Calculus BC Practice Quiz: Determining Absolute or Conditional Convergence

Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026

Test your understanding with short quizzes. This quiz has 10 questions to check your progress.

Question 1 of 10

According to the provided content, which of the following lists all possible classifications for a series?

All Questions (10)

According to the provided content, which of the following lists all possible classifications for a series?

A) Convergent or divergent

B) Absolutely convergent or conditionally convergent

C) Absolutely convergent, conditionally convergent, or divergent

D) Geometric, telescoping, or alternating

Correct Answer: C

The provided content explicitly states: 'A series may be absolutely convergent, conditionally convergent, or divergent.'

If the series of absolute values, Σ|a_n|, is known to converge, what can be concluded about the original series, Σa_n?

A) Σa_n must converge.

B) Σa_n must diverge.

C) Σa_n may either converge or diverge.

D) Σa_n must be conditionally convergent.

Correct Answer: A

The content includes the theorem: 'If a series converges absolutely, then it converges.' The convergence of Σ|a_n| means the series Σa_n is absolutely convergent, and therefore it must converge.

A series Σa_n is defined as conditionally convergent. Based on the provided content, which statement about the series Σa_n and the series Σ|a_n| must be true?

A) Both Σa_n and Σ|a_n| converge.

B) Σa_n converges, but Σ|a_n| diverges.

C) Σa_n diverges, but Σ|a_n| converges.

D) Both Σa_n and Σ|a_n| diverge.

Correct Answer: B

The classification 'conditionally convergent' applies to series that converge, but not absolutely. For a series to not be absolutely convergent, the series of its absolute values, Σ|a_n|, must diverge.

Under which condition is the sum of a convergent series guaranteed to remain unchanged if its terms are rearranged?

A) The series is conditionally convergent.

B) The series is absolutely convergent.

C) The series contains only positive terms.

D) The series is alternating.

Correct Answer: B

The content states, 'If a series converges absolutely, then any series obtained from it by regrouping or rearranging the terms has the same value.' This guarantee does not apply to conditionally convergent series.

If a series Σa_n is known to be divergent, what can be concluded about the series of its absolute values, Σ|a_n|?

A) Σ|a_n| must converge.

B) Σ|a_n| must diverge.

C) Σ|a_n| could either converge or diverge.

D) The sum of Σ|a_n| must be zero.

Correct Answer: B

This question tests the contrapositive of the theorem 'If a series converges absolutely, then it converges.' The contrapositive is 'If a series does not converge (i.e., diverges), then it does not converge absolutely.' For a series to not converge absolutely, Σ|a_n| must diverge.

A student has determined that a series Σb_n converges. To classify this convergence as either absolute or conditional, what is the next essential step?

A) Find the exact sum of Σb_n.

B) Determine if the terms b_n are positive.

C) Determine if the series Σ|b_n| converges or diverges.

D) Check if the terms of the series can be rearranged.

Correct Answer: C

The definition separating absolute and conditional convergence depends on the behavior of the series of absolute values. If Σ|b_n| converges, the original series is absolutely convergent. If Σ|b_n| diverges, it is conditionally convergent.

Which of the following statements is always true based on the provided content?

A) If a series converges, it must also converge absolutely.

B) If a series converges absolutely, it is also a convergent series.

C) If a series is conditionally convergent, its terms can be rearranged without changing the sum.

D) A series is either absolutely convergent or divergent.

Correct Answer: B

The content directly states the theorem: 'If a series converges absolutely, then it converges.' Option A is false (e.g., conditional convergence). Option C is false, as the rearrangement property is only guaranteed for absolutely convergent series. Option D is false as it omits conditional convergence.

It is discovered that a convergent series, Σc_n, can be rearranged to form a new series that converges to a different sum. What must be true about Σc_n?

A) Σc_n is divergent.

B) Σc_n is absolutely convergent.

C) Σc_n is not absolutely convergent.

D) No conclusion about its convergence type can be made.

Correct Answer: C

This is the contrapositive of the rearrangement theorem. The theorem states that if a series is absolutely convergent, its sum is invariant under rearrangement. Since the sum of Σc_n changes upon rearrangement, it cannot be absolutely convergent.

If a series is absolutely convergent, which of the following classifications is impossible for that series?

A) Convergent

B) Conditionally convergent

C) A series whose sum is preserved under rearrangement

D) A series of numbers

Correct Answer: B

The categories of absolutely convergent, conditionally convergent, and divergent are mutually exclusive. A series cannot be both absolutely convergent (meaning Σ|a_n| converges) and conditionally convergent (meaning Σ|a_n| diverges).

Consider the relationship between a series Σa_n and the series of its absolute values Σ|a_n|. Which statement correctly describes a one-way implication presented in the content?

A) The convergence of Σa_n implies the convergence of Σ|a_n|.

B) The divergence of Σ|a_n| implies the divergence of Σa_n.

C) The convergence of Σ|a_n| implies the convergence of Σa_n.

D) The divergence of Σa_n| implies the convergence of Σ|a_n|.

Correct Answer: C

The content provides the theorem: 'If a series converges absolutely, then it converges.' This means that the convergence of Σ|a_n| (absolute convergence) implies the convergence of Σa_n. Option A is not always true (the case of conditional convergence). Option B is not always true (the case of conditional convergence). Option D is logically incorrect.