Textbook Question
11–27. Alternating Series Test Determine whether the following series converge.
∑ (k = 0 to ∞) (−1)ᵏ / (2k + 1)
Ch. 10 - Sequences and Infinite Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Not the one you use?Change textbookSelect a different textbook
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 10.5.17
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) (−k)³ / (3k³ + 2)
Verified step by step guidance1
First, write down the general term of the series: \(a_k = \frac{(-k)^3}{3k^3 + 2}\).
Simplify the term \(a_k\): since \((-k)^3 = -k^3\), rewrite \(a_k\) as \(a_k = \frac{-k^3}{3k^3 + 2}\).
Analyze the behavior of \(a_k\) as \(k \to \infty\) by dividing numerator and denominator by \(k^3\): \(a_k = \frac{-1}{3 + \frac{2}{k^3}}\).
Determine the limit of \(a_k\) as \(k \to \infty\): \(\lim_{k \to \infty} a_k = \frac{-1}{3 + 0} = -\frac{1}{3}\).
Since the limit of \(a_k\) is not zero, conclude by the Test for Divergence (also called the nth-term test) that the series \(\sum_{k=1}^\infty a_k\) diverges.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Convergence of Infinite Series
An infinite series converges if the sequence of its partial sums approaches a finite limit. Understanding convergence is essential to determine whether the sum of infinitely many terms results in a finite value or diverges to infinity or oscillates without settling.
Recommended video:
06:52
Convergence of an Infinite Series
Limit Comparison Test
The Limit Comparison Test compares a given series with a known benchmark series by examining the limit of their term ratios. If the limit is a positive finite number, both series either converge or diverge together, helping to determine the behavior of complex series.
Recommended video:
07:45Limit Comparison Test
Behavior of Polynomial Terms in Series
When series terms involve polynomials, the dominant powers in numerator and denominator dictate the term's behavior as k approaches infinity. Simplifying these terms helps identify if the series resembles a p-series or another known type, aiding in applying appropriate convergence tests.
Recommended video:
07:00Taylor Polynomials
Related Practice
Textbook Question
9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.
∑ (k = 4 to ∞) (1 + cos²(k)) / (k − 3)
Textbook Question
9–30. The Ratio and Root Tests Use the Ratio Test or the Root Test to determine whether the following series converge absolutely or diverge.
∑ (from k = 1 to ∞) ((k / (k + 1)) × 2k²)
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 ∞) 3ᵏ⁺² / 5ᵏ
Textbook Question
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) (−7)ᵏ / k!
Textbook Question
45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (k = 1 to ∞) (−1)ᵏ · tan⁻¹(k) / k³