Question

Evaluate the limit of the following sequences. $$a_{n}=\frac{4^{n}+5 n !}{n !+2^{n}}$$

Short answer

Answer: As \(n \to \infty\), the limit of the sequence is 5.

Step-by-step solution

Observe the sequence expression

Let's take a closer look at the given sequence: $$a_n = \frac{4^n + 5n!}{n! + 2^n}$$ As n approaches infinity, we will compare the growth rates of the terms in the numerator and denominator.

Identify dominant terms in numerator and denominator

As n approaches infinity, the dominant term in the numerator is \(5n!\), as this term will grow much faster than \(4^n\). Similarly, the dominant term in the denominator is \(n!\), as this term will grow much faster than \(2^n\). We can rewrite our sequence to focus on the dominant terms: $$a_n \approx \frac{5n!}{n!}$$ To simplify, we can cancel \(n!\) in both the numerator and the denominator: $$a_n \approx 5$$ Since the other terms do not affect the limit significantly, we can now conclude:

Find the limit

As \(n \to \infty\): $$\lim_{n\to\infty} a_n = 5$$

Key concepts

Limit of a Sequence

A sequence is a list of numbers in a specific order. Understanding the limit of a sequence is crucial for analyzing its long-term behavior. The concept of a limit in sequences refers to what value the terms of the sequence get closer to as the index grows indefinitely. Consider the given sequence:\[a_n = \frac{4^n + 5n!}{n! + 2^n}\]To find the limit, we examine how the terms behave as \(n\) approaches infinity. If the terms stabilize to converge to a single value, that value is the limit. In this case, as \(n\) becomes very large, the sequence simplifies to a form where the factorial terms significantly overpower the exponential terms, simplifying the limit to \(5\). By simplifying the sequence appropriately, we find:\[\lim_{n\to\infty} a_n = 5\]Finding the limit of a sequence often involves identifying and focusing on dominant terms, which dictates the sequence's behavior at infinity.

Factorial Growth

Factorial growth is an essential concept when analyzing sequences. The factorial notation \(n!\) indicates the product of all positive integers up to \(n\). As \(n\) grows, \(n!\) grows exceptionally fast, even more so than exponential terms like \(2^n\) or \(4^n\).In our sequence, \(5n!\) in the numerator and \(n!\) in the denominator show how factorial growth dominates. Factorials grow super-exponentially, meaning they increase rapidly compared to simple powers. Understanding this growth helps identify the sequence's prominent terms, as was done in this case by comparing \(n!\) to other components. Recognizing \(n!\) as the dominant growth factor allows for simplifying the sequence effectively, leading to a clear evaluation of the sequence's limit.

Dominant Term Analysis

Dominant term analysis is a strategic approach to simplifying sequences. By focusing on terms with the highest growth rates, you can efficiently evaluate the overall behavior of complex sequences.For the given sequence:\[a_n = \frac{4^n + 5n!}{n! + 2^n}\]Identifying \(5n!\) as the dominant term in the numerator and \(n!\) as the dominant term in the denominator simplifies our task. This analysis highlights that these terms will eventually "overpower" other terms as \(n\) increases.With dominant terms:- Numerator: \(5n!\) surpasses \(4^n\)- Denominator: \(n!\) surpasses \(2^n\)Canceling out the \(n!\) terms results in a much simpler expression. Thus, dominant term analysis is invaluable for calculating sequence limits, as it reduces complexities and allows for straightforward limit evaluation.