Textbook Question
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{(n + 1)!⁄n!}
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 10.2.13
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{n³⁄(n⁴ + 1)}
Verified step by step guidance1
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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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.
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Guided course
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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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.
Recommended video:
6:04Intro to Rational Functions
Related Practice
Textbook Question
21–42. Geometric series Evaluate each geometric series or state that it diverges.
31.∑ (k = 1 to ∞) 2^(–3k)
Textbook Question
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 0 to ∞)3k / ∜(k⁴ + 3)
Textbook Question
Growth rates of sequences
Use Theorem 10.6 to find the limit of the following sequences or state that they diverge.
{n¹⁰ / ln20n}
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Textbook Question
Property of divergent series Prove Property 2 of Theorem 10.8: If ∑ aₖ diverges, then ∑ caₖ also diverges, for any real number c ≠ 0.
Textbook Question
Simplify k! / (k + 2)! for any integer k ≥ 0.
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