Textbook Question
35–44. Limits of sequences Write the terms a₁, a₂, a₃, and a₄ of the following sequences. If the sequence appears to converge, make a conjecture about its limit. If the sequence diverges, explain why.
aₙ = 1⁄10ⁿ; n = 1, 2, 3, …
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 10.5.19
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)
Verified step by step guidance1
Identify the given series: \( \sum_{k=4}^{\infty} \frac{1 + \cos^{2}(k)}{k - 3} \). We want to determine if this series converges or diverges.
Analyze the behavior of the terms for large \(k\). Notice that \(1 + \cos^{2}(k)\) oscillates between 1 and 2 because \(\cos^{2}(k)\) is always between 0 and 1.
Choose a comparison series to apply the Comparison Test or Limit Comparison Test. Since the denominator is \(k - 3\), which behaves like \(k\) for large \(k\), consider the series \( \sum_{k=4}^{\infty} \frac{1}{k} \), which is a harmonic series known to diverge.
Apply the Limit Comparison Test by computing the limit \( L = \lim_{k \to \infty} \frac{\frac{1 + \cos^{2}(k)}{k - 3}}{\frac{1}{k}} = \lim_{k \to \infty} \frac{(1 + \cos^{2}(k)) \cdot k}{k - 3} \). Simplify this expression to understand the behavior of \(L\).
Since \(1 + \cos^{2}(k)\) oscillates but stays bounded between 1 and 2, the limit \(L\) will be a finite positive number. By the Limit Comparison Test, the original series behaves like the harmonic series \(\sum \frac{1}{k}\), which diverges. Therefore, conclude about the convergence or divergence of the original series.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Comparison Test
The Comparison Test determines the convergence of a series by comparing it to a second series with known behavior. If the terms of the given series are less than or equal to the terms of a convergent series, it also converges. Conversely, if the terms are greater than or equal to those of a divergent series, it diverges.
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09:25
Direct Comparison Test
Limit Comparison Test
The Limit Comparison Test compares two series by taking the limit of the ratio of their terms. If this limit is a positive finite number, both series either converge or diverge together. This test is useful when direct comparison is difficult but the series have similar term behavior for large indices.
Recommended video:
07:45Limit Comparison Test
Behavior of Trigonometric Functions in Series
Understanding the bounded nature of trigonometric functions like cosine is crucial. Since cos²(k) oscillates between 0 and 1, the numerator (1 + cos²(k)) stays between 1 and 2, allowing simplification in comparison tests. Recognizing this helps in estimating the series terms and choosing an appropriate comparison series.
Recommended video:
6:04Introduction to Trigonometric Functions
Related Practice
Textbook Question
32–49. Choose your test Use the test of your choice to determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (from k = 1 to ∞) (−1)ᵏ k³ / √(k⁸ + 1)
Textbook Question
11–27. Alternating Series Test Determine whether the following series converge.
∑ (k = 0 to ∞) (−1)ᵏ / (2k + 1)
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
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) (−k)³ / (3k³ + 2)
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