Sequences and Series Practice Questions

AP (Advanced Placement) · AP Calculus BC · 151 free MCQs with instant results and detailed explanations.

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Sample Questions from Sequences and Series

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Easy

Which of the following sequences is increasing?

A. a_n = n^2

B. b_n = (-1)^n

C. c_n = 1/n

D. d_n = n*(-1)^n

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Correct Answer: A

The sequence a_n = n^2 is increasing because as n increases, n^2 also increases. The other sequences either oscillate or decrease.

Easy

Which of the following represents a convergent series?

A. Σ (1/n^2) from n=1 to ∞

B. Σ (1/n) from n=1 to ∞

C. Σ (1/n^3) from n=1 to ∞

D. Both A and C

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Correct Answer: D

Both series Σ (1/n^2) and Σ (1/n^3) converge by the p-series test, where p > 1 indicates convergence. Series Σ (1/n) diverges.

Easy

What is the limit of the sequence defined by a_n = 1/n as n approaches infinity?

A. 0

B. 1

C. Infinity

D. Does not exist

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Correct Answer: A

As n increases, the value of 1/n decreases and approaches 0. Therefore, the limit of the sequence is 0.

Medium

Which of the following series converges?

A. ∑ (1/n^2) from n=1 to ∞

B. ∑ (1/n) from n=1 to ∞

C. ∑ (3/n) from n=1 to ∞

D. ∑ (n^2/n^3) from n=1 to ∞

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Correct Answer: A

The series ∑ (1/n^2) converges by the p-series test where p=2 > 1. The other series diverge.

Medium

What is the limit of the sequence defined by a_n = (2^n + 3^n) / (5^n) as n approaches infinity?

A. 0

B. 1

C. 2

D. 3

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Correct Answer: A

As n approaches infinity, 3^n dominates in the numerator, and dividing both terms by 5^n gives the limit as 0.

Medium

Evaluate the convergence of the series ∑ (n!/n^n) from n=1 to ∞ using the ratio test.

A. Converges

B. Diverges

C. Conditionally converges

D. Cannot be determined

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Correct Answer: A

Applying the ratio test, the limit approaches 0, indicating that the series converges.

Medium

For the series ∑ (1/(n^2 + k)) where k is a positive constant, how does k affect convergence?

A. Only influences the sum but not convergence

B. Makes the series diverge for all k > 0

C. Only affects convergence for k = 0

D. Converges for any k > 0

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Correct Answer: D

For k > 0, the series behaves like a p-series with p=2, which converges regardless of k.

Hard

Consider the series ∑ (n=1 to ∞) (1/n^2). Which of the following statements about this series is true?

A. The series converges to π^2/6.

B. The series diverges to infinity.

C. The series converges to e.

D. The series diverges but oscillates between values.

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Correct Answer: A

The series ∑ (1/n^2) converges, and it is well-known that its sum equals π^2/6. This is a classic result in calculus, derived from the Basel problem.

Hard

Evaluate the limit of the sequence defined by a_n = (n^3 + 2n) / (3n^3 + 5). What is the limit as n approaches infinity?

A. 1/3

B. 0

C. 1

D. 2/3

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Correct Answer: A

To find the limit as n approaches infinity, divide each term by n^3. The limit simplifies to (1 + 0) / (3 + 0), which equals 1/3.

Q10

Hard

Given the series ∑(n=1 to ∞) (1/n^2), what is the value of the series?

A. π²/6

B. 1/2

C. 1

D. e

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Correct Answer: A

The series is known as the Basel problem, and it converges to π²/6, which is a well-established result in mathematical analysis.

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Sequences and Series — AP (Advanced Placement) AP Calculus BC Practice Questions Online

This page contains 151 practice MCQs for the chapter Sequences and Series in AP (Advanced Placement) AP Calculus BC. The questions are organized by difficulty — 39 easy, 86 medium, 26 hard — so you can choose the right level for your preparation.

Every question includes a detailed explanation to help you understand the concept, not just memorize answers. Take a timed quiz to simulate exam conditions, or practice at your own pace with no time limit.