Textbook Question
Use the Integral Test to show that the series
∑ (from n=0 to ∞) e^(−n²)
converges.
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 10.1.36
Convergence and Divergence
Which of the sequences {aₙ} in Exercises 31–100 converge, and which diverge? Find the limit of each convergent sequence.
aₙ = (n + 3) / (n² + 5n + 6)
Verified step by step guidance1
Identify the given sequence: \(a_n = \frac{n + 3}{n^2 + 5n + 6}\).
To analyze convergence or divergence, examine the behavior of \(a_n\) as \(n\) approaches infinity, i.e., find \(\lim_{n \to \infty} a_n\).
Since the sequence is a rational function of \(n\), compare the degrees of the numerator and denominator polynomials. The numerator is degree 1, and the denominator is degree 2.
Divide both numerator and denominator by the highest power of \(n\) in the denominator, which is \(n^2\), to simplify the limit expression: \(a_n = \frac{\frac{n}{n^2} + \frac{3}{n^2}}{\frac{n^2}{n^2} + \frac{5n}{n^2} + \frac{6}{n^2}} = \frac{\frac{1}{n} + \frac{3}{n^2}}{1 + \frac{5}{n} + \frac{6}{n^2}}\).
Evaluate the limit by letting \(n\) approach infinity, noting that terms with \(\frac{1}{n}\) and \(\frac{1}{n^2}\) approach zero, and conclude whether the sequence converges or diverges based on this limit.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Sequence Convergence and Divergence
A sequence converges if its terms approach a specific finite value as n approaches infinity; otherwise, it diverges. Understanding this helps determine the behavior of {aₙ} as n grows large.
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8:22
Introduction to Sequences
Limits of Sequences
The limit of a sequence is the value that the terms get arbitrarily close to as n becomes very large. Calculating limits often involves simplifying expressions and applying limit laws.
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8:22Introduction to Sequences
Asymptotic Behavior of Rational Functions
For sequences defined by rational functions like (n + 3)/(n² + 5n + 6), analyzing the degrees of numerator and denominator polynomials helps determine the limit, since higher-degree denominators typically drive the sequence to zero.
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5:50Asymptotes of Hyperbolas
Related Practice
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]
Textbook Question
Using the Root Test
In Exercises 9–16, use the Root Test to determine if each series converges absolutely or diverges.
∑(from n=1 to ∞) [4ⁿ / (3n)ⁿ]
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 = nπ cos(nπ)
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Textbook Question
Uniqueness of limits Prove that limits of sequences are unique. That is, show that if L₁ and L₂ are numbers such that aₙ → L₁ and aₙ → L₂, then L₁ = L₂.
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Textbook Question
Find the value of b for which
1 + eᵇ + e²ᵇ + e³ᵇ + ⋯ = 9.