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 ∞) [(-1)ⁿ n² e⁻ⁿ]
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 10.6.1
Determining Convergence or Divergence
In Exercises 1–14, determine whether the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test.
∑ (from n = 1 to ∞) [(-1)ⁿ⁺¹ (1 / n^(3/2))]
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Identify the general term of the series: \(a_n = \frac{1}{n^{3/2}}\) and the alternating factor is \((-1)^{n+1}\), which means the series is alternating in sign.
Check the conditions of the Alternating Series Test (Leibniz Test): (1) The terms \(a_n\) must be positive, (2) \(a_n\) must be decreasing, and (3) \(\lim_{n \to \infty} a_n = 0\).
Verify that \(a_n = \frac{1}{n^{3/2}}\) is positive for all \(n\) and that it decreases as \(n\) increases because the denominator grows larger.
Calculate the limit \(\lim_{n \to \infty} \frac{1}{n^{3/2}}\) to confirm it approaches zero, satisfying the third condition of the Alternating Series Test.
Since all conditions are met, conclude that the alternating series converges by the Alternating Series 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 convergence of series whose terms alternate in sign. It requires that the absolute value of 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
Absolute vs Conditional Convergence
A series converges absolutely if the series of absolute values converges. If the original alternating series converges but not absolutely, it is conditionally convergent. Understanding this distinction helps classify the nature of convergence.
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07:51Choosing a Convergence Test
p-Series and Convergence
A p-series has the form ∑ 1/n^p and converges if p > 1. In this problem, the terms involve 1/n^(3/2), where p = 3/2 > 1, indicating the absolute value series converges, which implies absolute convergence of the alternating series.
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04:30P-Series and Harmonic Series
Related Practice
Textbook Question
Finding Taylor Polynomials
In Exercises 1–10, find the Taylor polynomials of orders 0, 1, 2, and 3 generated by f at a.
f(x) = √(1 − x),a = 0
Textbook Question
Convergence and Divergence
Which of the sequences {aₙ} in Exercises 31–100 converge, and which diverge? Find the limit of each convergent sequence.
aₙ = 2 + (0.1)ⁿ
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Textbook Question
Convergence or Divergence
Which of the series in Exercises 57–64 converge, and which diverge? Give reasons for your answers.
∑ (from n = 1 to ∞) [(-1)ⁿ (n!)ⁿ] / [n^(n²)]
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Textbook Question
In Exercises 53–56, determine how many terms should be used to estimate the sum of the entire series with an error of less than 0.001.
∑ (from n = 1 to ∞) [(-1)ⁿ⁺¹ (n / (n² + 1))]
Textbook Question
In Exercises 67–72, use the results of Exercises 63 and 64 to determine if each series converges or diverges.
∑(from n=2 to ∞) [(ln n)¹⁰⁰⁰ / n¹.⁰⁰¹]
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