Textbook Question
Power Series
In Exercises 47–56, (a) find the series’ radius and interval of convergence. Then identify the values of x for which the series converges (b) absolutely and (c) conditionally.
∑ (from n = 1 to ∞) (x + 4)ⁿ/(n3ⁿ)
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.PE.4
Determining Convergence of Sequences
Which of the sequences whose nth terms appear in Exercises 1–18 converge, and which diverge? Find the limit of each convergent sequence.
aₙ = 1 + (0.9)ⁿ
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Identify the general term of the sequence: \(a_n = 1 + (0.9)^n\).
Recall that a sequence converges if its terms approach a finite limit as \(n\) approaches infinity.
Analyze the behavior of the term \((0.9)^n\) as \(n \to \infty\). Since \(0.9\) is between \(-1\) and \(1\), \((0.9)^n\) approaches \(0\).
Use the limit laws to find the limit of \(a_n\): \(\lim_{n \to \infty} a_n = \lim_{n \to \infty} \left(1 + (0.9)^n\right) = 1 + 0\).
Conclude that the sequence converges and its limit is \(1\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definition of Sequence Convergence
A sequence converges if its terms approach a specific finite value, called the limit, as n approaches infinity. Formally, a sequence {aₙ} converges to L if for every small positive number ε, there exists an N such that for all n > N, |aₙ - L| < ε.
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Introduction to Sequences
Limits of Exponential Terms
When a sequence includes terms like (r)ⁿ where |r| < 1, these terms approach zero as n becomes very large. This property helps determine the limit of sequences involving exponential decay factors, such as (0.9)ⁿ approaching 0.
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Evaluating Limits of Sequences
To find the limit of a sequence, analyze each component separately and use limit laws. For example, in aₙ = 1 + (0.9)ⁿ, since (0.9)ⁿ → 0, the sequence converges to 1 + 0 = 1.
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Related Practice
Textbook Question
Power Series
In Exercises 47–56, (a) find the series’ radius and interval of convergence. Then identify the values of x for which the series converges (b) absolutely and (c) conditionally.
∑ (from n = 1 to ∞) xⁿ/nⁿ
Textbook Question
Power Series
In Exercises 47–56, (a) find the series’ radius and interval of convergence. Then identify the values of x for which the series converges (b) absolutely and (c) conditionally.
∑ (from n = 1 to ∞) (csch n)xⁿ
Textbook Question
Determining Convergence of Sequences
Which of the sequences whose nth terms appear in Exercises 1–18 converge, and which diverge? Find the limit of each convergent sequence.
aₙ = (-4)ⁿ/n!
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Textbook Question
Determining Convergence of Series
Which of the series in Exercises 25–44 converge absolutely, which converge conditionally, and which diverge? Give reasons for your answers.
∑ (from n = 2 to ∞) 1/[n(ln n)²]
Textbook Question
Maclaurin Series
Find Taylor series at x = 0 for the functions in Exercises 63–70.
cos (x³/√5)