AP - Calculus BC:: Infinite Sequences and Series Flashcards (50 Cards) — Study Online

Q: For what values of the common ratio \(r\) does the geometric series \(\sum ar^n\) converge? - **A)** \(|r| > 1\) - **B)** All values of \(r\) produce a convergent series - **C)** \(|r| < 1\) - **D)** \(r\) must be a positive integer

**Answer: C)** A geometric series converges exactly when successive terms shrink toward zero, requiring the ratio's magnitude to be strictly less than 1.

Q: Find the sum of the geometric series \(\sum_{n=0}^\infty 3\left(\dfrac{1}{2}\right)^n\). - **A)** \(1.5\) - **B)** \(\infty\) - **C)** \(6\) - **D)** \(3\)

**Answer: C)** \(S = \dfrac{a}{1-r} = \dfrac{3}{1-1/2} = \dfrac{3}{1/2} = 6\).

Q: The nth-Term Test for Divergence states that if \(\lim_{n \to \infty} a_n \ne 0\) (or the limit doesn't exist), then: - **A)** \(\sum a_n\) must CONVERGE - **B)** The series must be geometric - **C)** No conclusion can be drawn about convergence - **D)** \(\sum a_n\) must DIVERGE

**Answer: D)** If the individual terms don't shrink to zero, the partial sums cannot settle to a finite limit -- this test can only prove divergence, never convergence.

Curriculum Overview

Topics & Key Concepts

Ap Calculus

Sample Flashcard Questions & Answers

Showing 8 of 50 cards

Question #1 Active Recall

An infinite series \(\sum_{n=1}^\infty a_n\) is said to CONVERGE if:

- **A)** The terms \(a_n\) are all positive
- **B)** Every individual term \(a_n\) equals zero
- **C)** The sequence of partial sums \(S_N = \sum_{n=1}^N a_n\) approaches a finite limit as \(N \to \infty\)
- **D)** The series has only finitely many nonzero terms

Answer & Explanation:

**Answer: C)**

Convergence of a series is defined entirely in terms of whether its partial sums settle down to a specific finite value as more terms are added.

Question #2 Active Recall

The formula for the sum of an infinite geometric series \(\sum_{n=0}^\infty ar^n\) (with \(|r|

Answer & Explanation:

**Answer: B)**

This is the standard geometric series sum formula, valid precisely when the common ratio's absolute value is less than 1.

Question #3 Active Recall

For what values of the common ratio \(r\) does the geometric series \(\sum ar^n\) converge?

- **A)** \(|r| > 1\)
- **B)** All values of \(r\) produce a convergent series
- **C)** \(|r| < 1\)
- **D)** \(r\) must be a positive integer

Answer & Explanation:

**Answer: C)**

A geometric series converges exactly when successive terms shrink toward zero, requiring the ratio's magnitude to be strictly less than 1.

Question #4 Active Recall

Find the sum of the geometric series \(\sum_{n=0}^\infty 3\left(\dfrac{1}{2}\right)^n\).

- **A)** \(1.5\)
- **B)** \(\infty\)
- **C)** \(6\)
- **D)** \(3\)

Answer & Explanation:

**Answer: C)**

\(S = \dfrac{a}{1-r} = \dfrac{3}{1-1/2} = \dfrac{3}{1/2} = 6\).

Question #5 Active Recall

The nth-Term Test for Divergence states that if \(\lim_{n \to \infty} a_n \ne 0\) (or the limit doesn't exist), then:

- **A)** \(\sum a_n\) must CONVERGE
- **B)** The series must be geometric
- **C)** No conclusion can be drawn about convergence
- **D)** \(\sum a_n\) must DIVERGE

Answer & Explanation:

**Answer: D)**

If the individual terms don't shrink to zero, the partial sums cannot settle to a finite limit -- this test can only prove divergence, never convergence.

Question #6 Active Recall

Use the nth-Term Test to determine whether \(\sum_{n=1}^\infty \dfrac{n}{n+1}\) converges or diverges.

- **A)** Converges to \(0\)
- **B)** Converges to \(1\)
- **C)** Diverges, since \(\lim_{n \to \infty} \dfrac{n}{n+1} = 1 \ne 0\)
- **D)** The nth-Term Test cannot be applied here

Answer & Explanation:

**Answer: C)**

Since the terms approach \(1\) (not \(0\)), the nth-Term Test immediately confirms divergence.

Question #7 Active Recall

Why does the harmonic series \(\sum_{n=1}^\infty \dfrac{1}{n}\) diverge, even though its individual terms \(1/n\) approach zero?

- **A)** The terms of the harmonic series do not approach zero
- **B)** The terms shrink too slowly for the partial sums to level off; this shows the nth-Term Test's converse is false -- terms approaching zero is necessary but NOT sufficient for convergence
- **C)** The harmonic series is a geometric series with \(r=1\)
- **D)** The harmonic series actually converges

Answer & Explanation:

**Answer: B)**

The harmonic series is the classic counterexample showing that having terms shrink to zero does NOT guarantee convergence -- it's a famous, important exception every calculus student should know.

Question #8 Active Recall

A p-series \(\sum_{n=1}^\infty \dfrac{1}{n^p}\) converges if and only if:

- **A)** \(p\) is a positive integer
- **B)** \(p = 1\)
- **C)** \(p < 1\)
- **D)** \(p > 1\)

Answer & Explanation:

**Answer: D)**

The p-series test gives a clean dividing line: convergence requires the exponent to exceed 1 (the harmonic series, \(p=1\), is the boundary case that diverges).

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