Question

Evaluate the limit of the following sequences or state that the limit does not exist. $$a_{n}=\frac{4^{n}+5 n !}{n !+2^{n}}$$

Short answer

Answer: The limit does not exist.

Step-by-step solution

Examine the expression and write the limit

We want to find the limit of the sequence \(a_n\) as \(n\) approaches infinity, which can be written as: $$\lim_{n \to \infty} \frac{4^{n}+5 n !}{n !+2^{n}}$$

Simplify the expression using algebra and factorials

Observe that both the numerator and the denominator have factorials in them. We can factor out the factorial from the entire expression to simplify it: $$\lim_{n \to \infty} \frac{n!(\frac{4^{n}}{n!}+5)}{n!(1+\frac{2^{n}}{n!})}$$

Divide the expression by n!

Now, since both the numerator and the denominator have a common factor of \(n!\), we can cancel them out to simplify the expression further: $$\lim_{n \to \infty} \frac{\frac{4^{n}}{n!}+5}{1+\frac{2^{n}}{n!}}$$

Analyze the behavior of the remaining terms as n approaches infinity

As \(n\) approaches infinity, the terms \(\frac{4^{n}}{n!}\) and \(\frac{2^{n}}{n!}\) will have different behavior: - The term \(\frac{4^{n}}{n!}\) will approach infinity faster than the term \(\frac{2^{n}}{n!}\) because the numerator is exponential with a higher base. - The term \(\frac{2^{n}}{n!}\) will approach zero since the factorial in the denominator grows faster than the exponential in the numerator.

Replace the term that approaches zero and calculate the limit

We can rewrite the limit, replacing the term that approaches zero: $$\lim_{n \to \infty} \frac{\frac{4^{n}}{n!}+5}{1+0}$$ Now the limit simplifies as follows: $$\lim_{n \to \infty} \frac{4^{n}}{n!}+5$$ However, this limit does not exist because the term \(\frac{4^{n}}{n!}\) will still approach infinity as \(n\) approaches infinity.

Key concepts

Factorials in Limits

Factorials, denoted by the symbol !, represent the product of all positive integers less than or equal to a given number . In calculus, understanding how factorials behave in limits is crucial, as they often appear in the analysis of the growth of sequences and series.

Factorials grow extremely rapidly. For instance, 5! (read as "five factorial") is calculated as 5 × 4 × 3 × 2 × 1, resulting in 120.

While this growth is immense, it is always important to compare it with other types of growth, such as exponential growth, to examine their relationships as n tends to infinity.

When factorials are part of a sequence under a limit, they can be manipulated algebraically to simplify expressions. By factoring out n! from both the numerator and the denominator, we can often simplify the sequence to a form that is easier to evaluate as n goes to infinity.

Exponential Growth

Exponential growth is a pattern where quantities increase rapidly as a function of power of a base number. In terms of sequences, this is often represented by terms like 4^n or 2^n, where the base (4 or 2 in this case) is raised to the power of a variable n.

The key feature of exponential growth is how quickly it escalates compared to other types of growth, such as polynomial or factorial growth. For example, consider sequences like 4^n; they grow faster than numerical sequences with the same base, regardless of coefficients adjusted by factorials that often taper this sharp increase.

In the case of the sequence we analyzed, the term 4^n will dominate as n grows large because the power to which 4 is raised becomes substantially larger than the incremental increases by operations like factorial. Understanding this behavior is essential in predicting the long-term behavior of the sequence when it's subjected to infinity in limits.

Calculus Sequences

Calculus sequences are an integral part of mathematical analysis. They represent ordered lists of numbers following a particular rule or pattern that can be studied over infinite terms.

In calculus, exploring the limit of a sequence is a fundamental concept. The limit can reveal stable long-term behavior or apparent divergence. It helps identify where a sequence converges to a specific value or if it grows unboundedly.

Analyzing a sequence like \(a_n = \frac{4^n + 5n!}{n! + 2^n}\) involves examining the components involved, such as factorial and exponential terms. By isolating these parts, canceling common factors, and evaluating their growth rates, we can determine the sequence's behavior as napproaches infinity.

The combination of techniques used in determining limits helps us understand how sequences behave and if they converge to a limit, if they remain stable, or if they diverge indefinitely.