Textbook Question
Determining Convergence or Divergence
In Exercises 17–46, use any method to determine whether the series converges or diverges. Give reasons for your answer.
∑(from n=1 to ∞) [(2 + (−1)ⁿ) / 1.25ⁿ]
Ch. 10 - Infinite Sequences and Series
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 10, Problem 10.4.20
Determining Convergence or Divergence
Which of the series in Exercises 17–56 converge, and which diverge? Use any method, and give reasons for your answers.
∑ (from n=1 to ∞) (1 + cos n) / n²
Verified step by step guidance1
Identify the general term of the series: \(a_n = \frac{1 + \cos n}{n^2}\).
Recall that \(\cos n\) oscillates between \(-1\) and \(1\), so \(1 + \cos n\) is bounded between \(0\) and \(2\).
Since \(a_n\) behaves roughly like \(\frac{\text{bounded term}}{n^2}\), compare it to the convergent p-series \(\sum \frac{1}{n^2}\), where \(p=2 > 1\).
Apply the Comparison Test: because \(0 \leq a_n \leq \frac{2}{n^2}\) and \(\sum \frac{2}{n^2}\) converges, the original series converges by the Comparison Test.
Conclude that the series \(\sum_{n=1}^\infty \frac{1 + \cos n}{n^2}\) converges absolutely, since the absolute value \(|a_n| \leq \frac{2}{n^2}\) also forms a convergent p-series.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Convergence and Divergence of Infinite Series
An infinite series converges if the sum of its terms approaches a finite limit as the number of terms grows indefinitely. If the sum does not approach a finite value, the series diverges. Understanding this distinction is fundamental to analyzing series behavior.
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Convergence of an Infinite Series
Comparison Test for Series
The Comparison Test involves comparing a given series to a second series whose convergence behavior is known. If the terms of the given series are smaller than those of a convergent series, it also converges; if larger than a divergent series, it diverges. This test helps determine convergence by bounding.
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09:25Direct Comparison Test
Behavior of Trigonometric Functions in Series
Trigonometric functions like cosine oscillate between fixed bounds, affecting the terms of a series. When combined with a decreasing denominator (like n²), the oscillations are controlled, often allowing convergence tests to focus on the dominant term's behavior.
Recommended video:
6:04Introduction to Trigonometric Functions
Related Practice
Textbook Question
In Exercises 57–82, use any method to determine whether the series converges or diverges. Give reasons for your answer.
∑ (from n = 0 to ∞) [(2n + 3)(2ⁿ + 3) / (3ⁿ + 2)]
Textbook Question
In Exercises 57–82, use any method to determine whether the series converges or diverges. Give reasons for your answer.
∑ (from n = 2 to ∞) [(3n)! / (n!)³]
Textbook Question
Determining Convergence or Divergence
Which of the series in Exercises 17–56 converge, and which diverge? Use any method, and give reasons for your answers.
∑ (from n=1 to ∞) 1 / (1 + 2² + 3² + ⋯ + n²)
Textbook Question
Determining Convergence or Divergence
In Exercises 17–46, use any method to determine whether the series converges or diverges. Give reasons for your answer.
∑(from n=3 to ∞) [2n² / n²ⁿ]
Textbook Question
Find the first four nonzero terms in the Maclaurin series for the functions in Exercises 31–38.
(tan⁻¹x)²