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 ∞) [(x + 1)²ⁿ] / 9ⁿ
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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 10, Problem 10.1.73
Convergence and Divergence
Which of the sequences {aₙ} in Exercises 31–100 converge, and which diverge? Find the limit of each convergent sequence.
aₙ = (2n + 2)! / (2n − 1)!
Verified step by step guidance1
Identify the given sequence: \(a_n = \frac{(2n + 2)!}{(2n - 1)!}\).
Recall that factorial notation means the product of all positive integers up to that number, so \((2n + 2)!\) expands to \((2n + 2)(2n + 1)(2n)(2n - 1)!\).
Rewrite the sequence by factoring out \((2n - 1)!\) from the numerator to simplify the expression: \(a_n = \frac{(2n + 2)(2n + 1)(2n)(2n - 1)!}{(2n - 1)!}\).
Cancel the common factorial term \((2n - 1)!\) in numerator and denominator, leaving \(a_n = (2n + 2)(2n + 1)(2n)\).
Analyze the behavior of \(a_n\) as \(n \to \infty\): since it is a product of terms that grow without bound, the sequence diverges (does not converge to a finite limit).
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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 finite limit as n approaches infinity; otherwise, it diverges. Understanding this helps determine the behavior of {aₙ} by analyzing its limit or growth pattern.
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8:22
Introduction to Sequences
Factorials and Their Growth Rates
Factorials (n!) grow very rapidly as n increases. Comparing factorial expressions like (2n+2)! and (2n−1)! requires understanding how factorial terms expand and dominate, which is crucial for evaluating the limit of the sequence.
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5:22Factorials
Limit Evaluation Techniques for Sequences
Techniques such as simplifying factorial expressions, using ratio tests, or applying Stirling’s approximation help find limits of sequences involving factorials. These methods allow precise determination of whether the sequence converges and to what value.
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Related Practice
Textbook Question
Direct Comparison Test
In Exercises 1–8, use the Direct Comparison Test to determine if each series converges or diverges.
∑ (from n=1 to ∞) cos²n / n^(3/2)
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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 = 8^(1/n)
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Textbook Question
Absolute and Conditional Convergence
Which of the series in Exercises 15–48 converge absolutely, which converge, and which diverge? Give reasons for your answers.
∑ (from n = 1 to ∞) [(-1)ⁿ⁻¹ / (n² + 2n + 1)]
Textbook Question
Suppose that aₙ > 0 and limₙ→∞ n²aₙ = 0. Prove that ∑aₙ converges.
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Textbook Question
In Exercises 57–82, use any method to determine whether the series converges or diverges. Give reasons for your answer.
∑ (from n = 1 to ∞) [3ⁿ / n³]