Textbook Question
{Use of Tech} For what value of r does
∑ (k = 3 to ∞) r²ᵏ = 10?
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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 10.4.9
9–16. Divergence Test Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive.
∑ (k = 0 to ∞) k / (2k + 1)
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Identify the general term of the series: \(a_k = \frac{k}{2k + 1}\).
Recall the Divergence Test (also known as the nth-term test for divergence): if \(\lim_{k \to \infty} a_k \neq 0\), then the series \(\sum a_k\) diverges.
Calculate the limit of the general term as \(k\) approaches infinity: \(\lim_{k \to \infty} \frac{k}{2k + 1}\).
To find this limit, divide numerator and denominator by \(k\): \(\lim_{k \to \infty} \frac{1}{2 + \frac{1}{k}}\).
Evaluate the limit: as \(k \to \infty\), \(\frac{1}{k} \to 0\), so the limit becomes \(\frac{1}{2}\). Since this limit is not zero, by the Divergence Test, the series diverges.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Divergence Test
The Divergence Test states that if the limit of the terms of a series does not approach zero as k approaches infinity, then the series diverges. If the limit is zero, the test is inconclusive, and other methods must be used to determine convergence.
Recommended video:
05:44
Divergence Test (nth Term Test)
Limit of a Sequence
The limit of a sequence is the value that the terms of the sequence approach as the index k becomes very large. Evaluating this limit helps determine the behavior of the series' terms, which is essential for applying the Divergence Test.
Recommended video:
Guided course
8:22Introduction to Sequences
Infinite Series and Convergence
An infinite series is the sum of infinitely many terms. Understanding whether such a series converges (approaches a finite value) or diverges (does not approach a finite value) is fundamental in calculus, often requiring tests like the Divergence Test.
Recommended video:
06:52Convergence of an Infinite Series
Related Practice
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 ∞) ((-1)ᵏ⁺¹) × ((10k³ + k) / (9k³ + k + 1))ᵏ
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Textbook Question
What comparison series would you use with the Comparison Test to determine whether
∑ (k = 1 to ∞) 1 / (k² + 1) converges?
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Textbook Question
9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.
∑ (k = 1 to ∞) 1 / (k³ᐟ² + 1)
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Textbook Question
38–39. Examining a series two ways Determine whether the following series converge using either the Comparison Test or the Limit Comparison Test. Then use another method to check your answer.
39. ∑ (k = 1 to ∞) 1 / (k² + 2k + 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 ∞)tan⁻¹(1 / √k)
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