Textbook Question
11–27. Alternating Series Test Determine whether the following series converge.
∑ (k = 1 to ∞) (−1)ᵏ⁺¹ k² / (k³ + 1)
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Ch. 10 - Sequences and Infinite Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
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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07:51Choosing a Convergence Test
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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07:45Limit Comparison Test
Related Practice
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.
59. ∑ (k = –3 to ∞) 4 / ((4k – 3)(4k + 1))
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Textbook Question
33–38. {Use of Tech} Remainders in alternating series Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than 10⁻⁴ in magnitude. Although you do not need it, the exact value of the series is given in each case.
ln 2 = ∑ (k = 1 to ∞) (−1)ᵏ⁺¹ / k
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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 + 1 / (2k))ᵏ
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Textbook Question
Property of divergent series Prove Property 2 of Theorem 10.8: If ∑ aₖ diverges, then ∑ caₖ also diverges, for any real number c ≠ 0.
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Textbook Question
What comparison series would you use with the Comparison Test to determine whether ∑ (k = 1 to ∞) 2ᵏ / (3ᵏ + 1) converges?
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