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 10.6.80
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)]
Verified step by step guidance1
First, write down the general term of the series: \(a_n = \frac{(2n + 3)(2^n + 3)}{3^n + 2}\).
To determine convergence or divergence, analyze the behavior of \(a_n\) as \(n\) approaches infinity. Focus on the dominant terms in the numerator and denominator for large \(n\).
Approximate the dominant terms: in the numerator, \((2n + 3)(2^n + 3) \approx 2n \cdot 2^n = 2n \cdot 2^n\), and in the denominator, \(3^n + 2 \approx 3^n\) for large \(n\).
Simplify the approximate form of \(a_n\) for large \(n\): \(a_n \approx \frac{2n \cdot 2^n}{3^n} = 2n \left(\frac{2}{3}\right)^n\).
Use the Divergence Test (also called the nth-term test): if \(\lim_{n \to \infty} a_n \neq 0\), the series diverges. Otherwise, consider applying a convergence test such as the Ratio Test or Root Test to confirm convergence.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Infinite Series and Convergence
An infinite series is the sum of infinitely many terms. Determining whether a series converges means checking if the sum approaches a finite limit as the number of terms grows indefinitely. Understanding convergence is essential to analyze the behavior of the given series.
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06:52
Convergence of an Infinite Series
Comparison and Limit Comparison Tests
These tests help determine convergence by comparing the given series to a known benchmark series. The Comparison Test involves directly comparing terms, while the Limit Comparison Test uses the limit of the ratio of terms. They are useful when series involve exponential or polynomial expressions.
Recommended video:
07:45Limit Comparison Test
Ratio Test
The Ratio Test examines the limit of the absolute value of the ratio of consecutive terms. If this limit is less than one, the series converges absolutely; if greater than one, it diverges. This test is particularly effective for series with exponential terms like powers of n.
Recommended video:
05:35Ratio Test
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
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 + cos n) / n²
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
Find the value of b for which
1 + eᵇ + e²ᵇ + e³ᵇ + ⋯ = 9.