Textbook Question
"21–26. Recurrence relations Write the first four terms of the sequence {aₙ} defined by the following recurrence relations.
aₙ₊₁ = 3aₙ-12; a₁ = 10
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14. Sequences & Series
Sequences
2:30 minutesProblem 10.2.13
Textbook Question
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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Video duration:
2mKey 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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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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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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