Textbook Question
Error Estimates
The approximation eˣ = 1 + x + (x² / 2) is used when x is small. Use the Remainder Estimation Theorem to estimate the error when |x| < 0.1.
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 10.1.81
Convergence and Divergence
Which of the sequences {aₙ} in Exercises 31–100 converge, and which diverge? Find the limit of each convergent sequence.
aₙ = (xⁿ / (2n + 1))^(1/n),x > 0
Verified step by step guidance1
First, write down the given sequence explicitly: \(a_n = \left( \frac{x^n}{2n + 1} \right)^{\frac{1}{n}}\) where \(x > 0\).
Rewrite the sequence inside the \(n\)th root to separate the terms: \(a_n = \left( x^n \right)^{\frac{1}{n}} \cdot \left( \frac{1}{2n + 1} \right)^{\frac{1}{n}}\).
Simplify the powers: \(a_n = x \cdot \left( \frac{1}{2n + 1} \right)^{\frac{1}{n}}\).
Analyze the limit of the second factor as \(n \to \infty\): consider \(\lim_{n \to \infty} \left( \frac{1}{2n + 1} \right)^{\frac{1}{n}}\).
Use the fact that \(\lim_{n \to \infty} a_n^{1/n} = 1\) for any sequence \(a_n\) that tends to infinity, so this limit approaches 1. Therefore, the limit of \(a_n\) is \(x \cdot 1 = x\). Conclude that the sequence converges to \(x\).
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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 limit as n approaches infinity; otherwise, it diverges. Understanding this helps determine the behavior of {aₙ} by analyzing its limit. Convergence implies stability in the sequence's values, while divergence indicates unbounded or oscillatory behavior.
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Introduction to Sequences
Limit of a Sequence
The limit of a sequence {aₙ} is the value that the terms approach as n becomes very large. Calculating this limit often involves algebraic manipulation or applying limit laws. For the given sequence, finding the limit involves evaluating the nth root and the behavior of xⁿ and the denominator as n grows.
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nth Root and Exponential Growth
The nth root function, ( )^(1/n), moderates growth rates by reducing exponential terms. Since xⁿ grows exponentially, taking the nth root can simplify the expression to a form involving x. Understanding how the nth root interacts with exponential and polynomial terms is crucial to finding the sequence's limit.
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Related Practice
Textbook Question
Intervals of Convergence
In Exercises 1–36, (a) find the series’ radius and interval of convergence. For what values of x does the series converge (b) absolutely, (c) conditionally?
∑ (from n = 1 to ∞) [ (ln n) xⁿ ]
Textbook Question
Using the Ratio Test
In Exercises 1–8, use the Ratio Test to determine whether each series converges absolutely or diverges.
∑(from n=2 to ∞) [(3ⁿ⁺²) / ln(n)]
Textbook Question
Make up a geometric series ∑a rⁿ⁻¹ that converges to the number 5 if
b. a = 13/2
Textbook Question
In Exercises 43–50, use Theorem 20 to find the series’ interval of convergence and, within this interval, the sum of the series as a function of x.
∑ (from n = 0 to ∞) [ (x² − 1) / 2 ]ⁿ
Textbook Question
Make up an infinite series of nonzero terms whose sum is
b. −3