Textbook Question
Explain why or why not
Determine whether the following statements are true and give an explanation or counterexample.
g.The series ∑ (from k = 1 to ∞) (k² / (k² + 1)) converges.
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Ch. 10 - Sequences and Infinite Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 10, Problem 10.R.17
12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.
aₙ = 8ⁿ / n!
Verified step by step guidance1
Identify the sequence given: \(a_n = \frac{8^n}{n!}\), where \(n!\) denotes the factorial of \(n\).
Recall that factorial \(n!\) grows much faster than any exponential function \$8^n\( as \)n$ becomes very large.
To analyze the limit \(\lim_{n \to \infty} \frac{8^n}{n!}\), consider the behavior of the terms in the denominator compared to the numerator as \(n\) increases.
Use the ratio test for sequences by examining the ratio \(\frac{a_{n+1}}{a_n} = \frac{8^{n+1}/(n+1)!}{8^n/n!} = \frac{8}{n+1}\), which approaches 0 as \(n \to \infty\).
Since the ratio approaches 0, the terms \(a_n\) get smaller and smaller, indicating that the limit of the sequence is 0.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limits of Sequences
The limit of a sequence describes the value that the terms of the sequence approach as the index n becomes very large. If the terms get arbitrarily close to a specific number, the sequence converges to that limit; otherwise, it diverges.
Recommended video:
Guided course
8:22
Introduction to Sequences
Factorials and Growth Rates
Factorials (n!) grow much faster than exponential functions like 8ⁿ as n increases. Understanding this difference in growth rates helps determine the behavior of sequences involving factorials and exponentials.
Recommended video:
5:22Factorials
Ratio Test for Sequences
The ratio test compares the ratio of consecutive terms aₙ₊₁ / aₙ to analyze the limit of a sequence. If this ratio approaches a value less than 1, the sequence tends to zero, aiding in evaluating limits involving factorials and exponentials.
Recommended video:
05:35Ratio Test
Related Practice
Textbook Question
12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.
aₙ = (–1)ⁿ (3n³ + 4n) / (6n³ + 5)
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Textbook Question
42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.
∑ (from k = 1 to ∞)3 / (2 + eᵏ)
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Textbook Question
42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.
∑ (from k = 1 to ∞)2ᵏ / eᵏ
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Textbook Question
42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.
∑ (from k = 1 to ∞)(7 + sin k) / k²
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Textbook Question
77–87. Absolute or conditional convergence
Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (from k = 1 to ∞)(−1)ᵏ⁺¹(k² + 4) / (2k² + 1)
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