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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblem 10.6.51
Chapter 10, Problem 10.6.51

45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (k = 1 to ∞) cos(k) / k³

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Identify the given series: \( \sum_{k=1}^{\infty} \frac{\cos(k)}{k^3} \). We want to determine if it converges absolutely, conditionally, or diverges.
Recall that absolute convergence means the series \( \sum |a_k| \) converges, where \( a_k = \frac{\cos(k)}{k^3} \). So consider the series \( \sum_{k=1}^{\infty} \left| \frac{\cos(k)}{k^3} \right| = \sum_{k=1}^{\infty} \frac{|\cos(k)|}{k^3} \).
Since \( |\cos(k)| \leq 1 \) for all integers \( k \), the terms \( \frac{|\cos(k)|}{k^3} \) are bounded above by \( \frac{1}{k^3} \). We know that \( \sum_{k=1}^{\infty} \frac{1}{k^3} \) is a p-series with \( p = 3 > 1 \), which converges.
By the Comparison Test, since \( 0 \leq \frac{|\cos(k)|}{k^3} \leq \frac{1}{k^3} \) and \( \sum \frac{1}{k^3} \) converges, the series \( \sum \frac{|\cos(k)|}{k^3} \) converges. Therefore, the original series converges absolutely.
Since absolute convergence implies convergence, there is no need to check for conditional convergence. The series \( \sum_{k=1}^{\infty} \frac{\cos(k)}{k^3} \) converges absolutely.

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Key Concepts

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

Absolute Convergence

A series ∑a_k converges absolutely if the series of absolute values ∑|a_k| converges. Absolute convergence guarantees convergence regardless of the terms' signs, and it often simplifies analysis by allowing the use of comparison tests on positive terms.
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Choosing a Convergence Test

Conditional Convergence

A series converges conditionally if it converges, but does not converge absolutely. This means ∑a_k converges, but ∑|a_k| diverges. Conditional convergence often occurs in alternating or oscillating series where sign changes affect convergence.
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Comparison and Limit Comparison Tests

These tests help determine convergence by comparing a given series to a known benchmark series. For example, comparing ∑|cos(k)/k³| to ∑1/k³, a p-series with p=3, which converges, can establish absolute convergence of the original series.
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Related Practice
Textbook Question

Suppose the sequence { aₙ} is defined by the explicit formula aₙ = 1/n, for n=1, 2, 3, .....Write out the first five terms of the sequence.

Textbook Question

46–53. Decimal expansions

Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers).


49.0.037̅ = 0.037037…

Textbook Question

54–69. Telescoping series

For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sₙ}. Then evaluate limₙ→∞ Sₙ to obtain the value of the series or state that the series diverges.


57. ∑ (k = 1 to ∞) 1 / ((k + 6)(k + 7))

Textbook Question

48–63. Choose your test Determine whether the following series converge or diverge using the properties and tests introduced in Sections 10.3 and 10.4.

∑ (k = 1 to ∞) k / √(k² + 1)

Textbook Question

13–52. Limits of sequences

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


{(3n³ − 1)⁄(2n³ + 1)}

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Textbook Question

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.

∑ (from k = 1 to ∞) (−1)ᵏ × k / (k³ + 1)