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 guidance

1. Identify the given sequence: \(a_n = \frac{(n+1)!}{n!}\).
2. Recall the definition of factorial: \(n! = n \times (n-1) \times (n-2) \times \cdots \times 1\).
3. Simplify the expression by canceling common factorial terms: \(\frac{(n+1)!}{n!} = \frac{(n+1) \times n!}{n!} = n+1\).
4. Analyze the simplified sequence \(a_n = n+1\) as \(n\) approaches infinity.
5. Conclude whether the sequence converges or diverges based on the behavior of \(n+1\) as \(n \to \infty\).

Key Concepts

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.

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.

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.