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Chapter 8: Problem 82

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

Expert verified
Answer: The limit does not exist.

Step by step solution

01

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: $$\lim_{n \to \infty} \frac{4^{n}+5 n !}{n !+2^{n}}$$
02

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: $$\lim_{n \to \infty} \frac{n!(\frac{4^{n}}{n!}+5)}{n!(1+\frac{2^{n}}{n!})}$$
03

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: $$\lim_{n \to \infty} \frac{\frac{4^{n}}{n!}+5}{1+\frac{2^{n}}{n!}}$$
04

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.
05

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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 脳 4 脳 3 脳 2 脳 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.

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Most popular questions from this chapter

The Greek philosopher Zeno of Elea (who lived about 450 B.c.) invented many paradoxes, the most famous of which tells of a race between the swift warrior Achilles and a tortoise. Zeno argued The slower when running will never be overtaken by the quicker: for that which is pursuing must first reach the point from which that which is fleeing started, so that the slower must necessarily always be some distance ahead. In other words, by giving the tortoise a head start, Achilles will never overtake the tortoise because every time Achilles reaches the point where the tortoise was, the tortoise has moved ahead. Resolve this paradox by assuming that Achilles gives the tortoise a 1 -mi head start and runs \(5 \mathrm{mi} / \mathrm{hr}\) to the tortoise's \(1 \mathrm{mi} / \mathrm{hr}\). How far does Achilles run before he overtakes the tortoise, and how long does it take?

A ball is thrown upward to a height of \(h_{0}\) meters. After each bounce, the ball rebounds to a fraction r of its previous height. Let \(h_{n}\) be the height after the nth bounce and let \(S_{n}\) be the total distance the ball has traveled at the moment of the nth bounce. a. Find the first four terms of the sequence \(\left\\{S_{n}\right\\}\) b. Make a table of 20 terms of the sequence \(\left\\{S_{n}\right\\}\) and determine \(a\) plausible value for the limit of \(\left\\{S_{n}\right\\}\) $$h_{0}=20, r=0.75$$

$$\text {Evaluate each series or state that it diverges.}$$ $$\sum_{k=1}^{\infty} \frac{\pi^{k}}{e^{k+1}}$$

Pick two positive numbers \(a_{0}\) and \(b_{0}\) with \(a_{0}>b_{0},\) and write out the first few terms of the two sequences \(\left\\{a_{n}\right\\}\) and \(\left\\{b_{n}\right\\}:\) $$a_{n+1}=\frac{a_{n}+b_{n}}{2}, \quad b_{n+1}=\sqrt{a_{n} b_{n}}, \quad \text { for } n=0,1,2 \dots$$ (Recall that the arithmetic mean \(A=(p+q) / 2\) and the geometric mean \(G=\sqrt{p q}\) of two positive numbers \(p\) and \(q\) satisfy \(A \geq G.)\) a. Show that \(a_{n} > b_{n}\) for all \(n\). b. Show that \(\left\\{a_{n}\right\\}\) is a decreasing sequence and \(\left\\{b_{n}\right\\}\) is an increasing sequence. c. Conclude that \(\left\\{a_{n}\right\\}\) and \(\left\\{b_{n}\right\\}\) converge. d. Show that \(a_{n+1}-b_{n+1}<\left(a_{n}-b_{n}\right) / 2\) and conclude that \(\lim _{n \rightarrow \infty} a_{n}=\lim _{n \rightarrow \infty} b_{n} .\) The common value of these limits is called the arithmetic-geometric mean of \(a_{0}\) and \(b_{0},\) denoted \(\mathrm{AGM}\left(a_{0}, b_{0}\right)\). e. Estimate AGM(12,20). Estimate Gauss' constant \(1 / \mathrm{AGM}(1, \sqrt{2})\).

Consider the geometric series \(f(r)=\sum_{k=0}^{\infty} r^{k},\) where \(|r|<1\) a. Fill in the following table that shows the value of the series \(f(r)\) for various values of \(r\) $$\begin{array}{|c|c|c|c|c|c|c|c|c|c|} \hline r & -0.9 & -0.7 & -0.5 & -0.2 & 0 & 0.2 & 0.5 & 0.7 & 0.9 \\ \hline f(r) & & & & & & & & & \\ \hline \end{array}$$ b. Graph \(f,\) for \(|r|<1\) \text { c. Evaluate } \lim _{r \rightarrow 1^{-}} f(r) \text { and } \lim _{r \rightarrow-1^{+}} f(r)
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