Domanda del libro di testo
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) (4k)! / (k!)⁴
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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
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 10, Problema 10.2.13
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{n³⁄(n⁴ + 1)}
Guida verificata passo dopo passo1
Identify the given sequence: \(a_n = \frac{n^3}{n^4 + 1}\).
To find the limit as \(n\) approaches infinity, analyze the highest powers of \(n\) in the numerator and denominator.
Divide both the numerator and denominator by \(n^4\), the highest power in the denominator, to simplify the expression: \(a_n = \frac{\frac{n^3}{n^4}}{\frac{n^4}{n^4} + \frac{1}{n^4}} = \frac{\frac{1}{n}}{1 + \frac{1}{n^4}}\).
Evaluate the limit of each part as \(n \to \infty\): \(\frac{1}{n} \to 0\) and \(\frac{1}{n^4} \to 0\), so the expression simplifies to \(\frac{0}{1 + 0} = 0\).
Conclude that the limit of the sequence \(a_n\) as \(n\) approaches infinity 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
Dominant Term Analysis
When evaluating limits of sequences involving polynomials, focus on the highest degree terms in the numerator and denominator. These dominant terms determine the behavior of the sequence as n approaches infinity, simplifying the limit calculation.
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Percorso guidato
05:44Divergence Test (nth Term Test)
Behavior of Rational Functions at Infinity
For sequences defined by rational functions, the limit as n approaches infinity depends on the degrees of the numerator and denominator polynomials. If the numerator's degree is less, the limit is zero; if equal, the limit is the ratio of leading coefficients; if greater, the sequence diverges.
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6:04Intro to Rational Functions
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13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
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