Ch. 10 - Sequences and Infinite Series

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 10 - Sequences and Infinite Series

Problem 10.2.9

Chapter 10, Problem 10.2.9

6–9. Determine whether the following sequences converge or diverge, and state whether they are monotonic or whether they oscillate. Give the limit when the sequence converges.

{(−0.7)ⁿ}

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Identify the given sequence: \(a_n = (-0.7)^n\).

Recall that a sequence converges if its terms approach a specific finite value as \(n\) approaches infinity.

Since \(| -0.7 | = 0.7 < 1\), the terms \((-0.7)^n\) get closer to zero as \(n\) increases, so the sequence converges to 0.

Determine the behavior of the sequence: because the base is negative, the terms alternate in sign, causing the sequence to oscillate between positive and negative values.

Conclude that the sequence converges to 0 and oscillates, so it is not monotonic.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Sequence Convergence and Divergence

A sequence converges if its terms approach a specific finite value as n approaches infinity; otherwise, it diverges. Determining convergence involves analyzing the behavior of the general term for large n, often using limits.

Monotonicity of Sequences

A sequence is monotonic if it is either entirely non-increasing or non-decreasing. Identifying monotonicity helps understand the sequence's behavior and is useful in proving convergence.

Oscillating Sequences

An oscillating sequence alternates in sign or fluctuates without settling to a single value. Recognizing oscillation is important to distinguish between sequences that converge and those that do not.

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Related Practice

Textbook Question

40–62. Choose your test Use the test of your choice to determine whether the following series converge.

∑ (k = 1 to ∞) 1 / k^(1 + p),p > 0

Textbook Question

Define the remainder of an infinite series.

Textbook Question

13–52. Limits of sequences

Find the limit of the following sequences or determine that the sequence diverges.

{tan⁻¹(10n⁄(10n + 4))}

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Textbook Question

1–10. Choosing convergence tests Identify a convergence test for each series. If necessary, explain how to simplify or rewrite the series before applying the convergence test. You do not need to carry out the convergence test.

∑ (from k = 3 to ∞) (2k²) / (k² − k − 2)

Textbook Question

21–42. Geometric series Evaluate each geometric series or state that it diverges.

21.∑ (k = 0 to ∞) (1/4)ᵏ

Textbook Question

For what values of r does the sequence {rⁿ} converge? Diverge?

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6–9. Determine whether the following sequences converge or diverge, and state whether they are monotonic or whether they oscillate. Give the limit when the sequence converges.{(−0.7)ⁿ}

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Key Concepts

Sequence Convergence and Divergence

Monotonicity of Sequences

Oscillating Sequences

6–9. Determine whether the following sequences converge or diverge, and state whether they are monotonic or whether they oscillate. Give the limit when the sequence converges.

{(−0.7)ⁿ}