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 10.3.64
Use the Integral Test to show that the series
∑ (from n=0 to ∞) e^(−n²)
converges.
Verified step by step guidance1
Identify the function corresponding to the terms of the series: define \( f(x) = e^{-x^2} \). This function is positive and continuous for all \( x \geq 0 \).
Check if \( f(x) \) is decreasing for \( x \geq 0 \). To do this, compute the derivative \( f'(x) \) and verify that it is negative for \( x > 0 \).
Set up the improper integral \( \int_0^{\infty} e^{-x^2} \, dx \) to apply the Integral Test. This integral is known as the Gaussian integral (or related to it).
Determine whether the integral \( \int_0^{\infty} e^{-x^2} \, dx \) converges. Since this integral converges to a finite value, it implies the series converges by the Integral Test.
Conclude that because \( f(x) = e^{-x^2} \) is positive, continuous, decreasing for \( x \geq 0 \), and the integral of \( f(x) \) from 0 to infinity converges, the series \( \sum_{n=0}^{\infty} e^{-n^2} \) converges.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integral Test for Series Convergence
The Integral Test determines the convergence of an infinite series by comparing it to an improper integral. If the function corresponding to the series terms is positive, continuous, and decreasing, then the series converges if and only if the integral of the function from some point to infinity converges.
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07:51
Choosing a Convergence Test
Behavior of the Function e^(−n²)
The function e^(−n²) decreases rapidly as n increases because the exponent is negative and grows quadratically. This rapid decay suggests the terms approach zero quickly, which is a key factor in the convergence of the series.
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4:47The Number e
Improper Integrals and Their Convergence
An improper integral extends over an infinite interval or involves an unbounded integrand. To apply the Integral Test, one evaluates the integral of e^(−x²) from a finite point to infinity, which is known to converge due to the Gaussian integral properties.
Recommended video:
11:11Improper Integrals: Infinite Intervals
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
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 = (n + 3) / (n² + 5n + 6)
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Textbook Question
Find the value of b for which
1 + eᵇ + e²ᵇ + e³ᵇ + ⋯ = 9.