Textbook Question
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{(√(4n⁴ + 3n))⁄(8n² + 1)}
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14. Sequences & Series
Sequences
2:17 minutesProblem 10.2.33
Textbook Question
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{(n + 1)!⁄n!}
Verified step by step guidance1
Identify the given sequence: \(a_n = \frac{(n+1)!}{n!}\).
Recall the definition of factorial: \(n! = n \times (n-1) \times (n-2) \times \cdots \times 1\).
Simplify the expression by canceling common factorial terms: \(\frac{(n+1)!}{n!} = \frac{(n+1) \times n!}{n!} = n+1\).
Analyze the simplified sequence \(a_n = n+1\) as \(n\) approaches infinity.
Conclude whether the sequence converges or diverges based on the behavior of \(n+1\) as \(n \to \infty\).
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Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definition of a Sequence Limit
The limit of a sequence is the value that the terms of the sequence approach as the index n goes to infinity. If the terms get arbitrarily close to a finite number, the sequence converges; otherwise, it diverges.
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Guided course
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Introduction to Sequences
Factorials and Their Properties
A factorial, denoted n!, is the product of all positive integers up to n. Understanding how factorials grow and simplify, such as (n+1)! = (n+1) × n!, is essential for manipulating sequences involving factorial expressions.
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Behavior of Rational Expressions Involving Factorials
When evaluating limits of sequences with factorials in numerator and denominator, simplifying the expression often reveals growth rates. Recognizing that (n+1)!/n! simplifies to (n+1) helps determine whether the sequence diverges or converges.
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