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 10.5.10
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]
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Identify the general term of the series: \(a_n = \frac{4^n}{(3n)^n}\).
Recall the Root Test formula: compute \(L = \lim_{n \to \infty} \sqrt[n]{|a_n|}\).
Apply the nth root to the term: \(\sqrt[n]{|a_n|} = \sqrt[n]{\frac{4^n}{(3n)^n}} = \frac{\sqrt[n]{4^n}}{\sqrt[n]{(3n)^n}}\).
Simplify the nth roots: \(\frac{4}{3n}\).
Evaluate the limit as \(n\) approaches infinity: \(L = \lim_{n \to \infty} \frac{4}{3n}\). Use this value to determine convergence: if \(L < 1\), the series converges absolutely; if \(L > 1\), it diverges; if \(L = 1\), the test is inconclusive.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Root Test for Series Convergence
The Root Test determines the convergence of an infinite series by examining the nth root of the absolute value of its terms. Specifically, if the limit of the nth root of |a_n| as n approaches infinity is less than 1, the series converges absolutely; if greater than 1, it diverges; and if equal to 1, the test is inconclusive.
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Root Test
Absolute Convergence
A series converges absolutely if the series of the absolute values of its terms converges. Absolute convergence guarantees convergence regardless of the sign of terms, making it a stronger form of convergence and simplifying the analysis of series with alternating or complex terms.
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Evaluating Limits Involving nth Roots
To apply the Root Test, one must compute the limit of the nth root of the series terms. This often involves simplifying expressions with powers and roots, such as rewriting terms like (4^n / (3n)^n)^(1/n) into a more manageable form to find the limit as n approaches infinity.
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06:13Limits of Rational Functions with Radicals
Related Practice
Textbook Question
Recursively Defined Sequences
In Exercises 101–108, assume that each sequence converges and find its limit.
a₁ = 2,aₙ₊₁ = 72 / (1 + aₙ)
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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)
1
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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
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.