Textbook Question
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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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 10.6.17
11–27. Alternating Series Test Determine whether the following series converge.
∑ (k = 1 to ∞) (−1)ᵏ⁺¹ k² / (k³ + 1)
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Identify the general term of the series: \(a_k = \frac{k^2}{k^3 + 1}\) and note the alternating factor \((-1)^{k+1}\), which makes this an alternating series.
Recall the Alternating Series Test, which states that an alternating series \(\sum (-1)^{k} b_k\) converges if two conditions are met: (1) the sequence \(b_k\) is decreasing, and (2) \(\lim_{k \to \infty} b_k = 0\).
Set \(b_k = \frac{k^2}{k^3 + 1}\) (the absolute value of the terms) and check the limit as \(k\) approaches infinity: compute \(\lim_{k \to \infty} \frac{k^2}{k^3 + 1}\) by dividing numerator and denominator by \(k^3\).
Analyze whether \(b_k\) is decreasing for sufficiently large \(k\) by considering the behavior of the function \(f(k) = \frac{k^2}{k^3 + 1}\) or by comparing \(b_k\) and \(b_{k+1}\).
If both conditions of the Alternating Series Test are satisfied (limit zero and decreasing terms), conclude that the series converges; otherwise, it does not converge by this test.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Alternating Series Test
The Alternating Series Test determines the convergence of series whose terms alternate in sign. It requires that the absolute value of the terms decreases monotonically to zero. If these conditions hold, the series converges, even if it does not converge absolutely.
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Alternating Series Test
Limit of the Terms
For a series to converge, the terms must approach zero as k approaches infinity. Evaluating the limit of the general term helps verify this condition, which is essential for applying the Alternating Series Test or any convergence test.
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05:50One-Sided Limits
Behavior of Rational Functions at Infinity
Understanding how rational functions behave as k grows large helps simplify the terms of the series. For example, k²/(k³ + 1) behaves like k²/k³ = 1/k for large k, which aids in analyzing the limit and monotonicity of the terms.
Recommended video:
6:04Intro to Rational Functions
Related Practice
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
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞)(ln²k) / k³ᐟ²
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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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