Domanda del libro di testo
12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.
aₙ = ((3n² + 2n + 1) · sin(n)) / (4n³ + n) (Hint: Use the Squeeze Theorem.)
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Ch. 10 - Sequences and Infinite Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- 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
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Capitolo 10, Problema 10.R.17
12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.
aₙ = 8ⁿ / n!
Guida verificata passo dopo passo1
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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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.
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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.
Video consigliato:
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.
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Percorso guidato
05:35Ratio Test
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42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.
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12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.
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42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.
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Domanda del libro di testo
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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