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Chapter 9: Problem 22

Find the limit of the following sequences or determine that the limit does not exist. $$\left\\{\left(1+\frac{4}{n}\right)^{3 n}\right\\}$$

Short Answer

Expert verified
Answer: The limit of the given sequence does not exist because it approaches infinity.

Step by step solution

01

Rewrite the expression in a helpful form

We will rewrite the sequence as follows: $$\left\\{\left(1+\frac{4}{n}\right)^{3 n}\right\\} = \left\\{\left[\left(1+\frac{1}{n}\right)^n\right]^{4n}\right\\}$$ This rewrite makes it clear that we can use the limit $$(1+\frac{1}{n})^n \Rightarrow e$$.
02

Apply the limit

Now we will apply the limit as n approaches infinity to the sequence term. $$\lim_{n\to\infty}\left\{\left[\left(1+\frac{1}{n}\right)^n\right]^{4n}\right\}$$
03

Apply the limit laws

Using the limit laws (in particular, the limit of a product is the product of the limits), $$\lim_{n\to\infty}\left[\left(1+\frac{1}{n}\right)^n\right]^{4n} = \left[\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n\right]^{4n}$$
04

Use the known limit

Recall that, as n approaches infinity, the limit of the term inside the bracket is e: $$\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n = e$$ So, our expression becomes: $$\left(e\right)^{4n}$$
05

Find the final limit

As n approaches infinity, $$\lim_{n\to\infty}\left(e\right)^{4n} = \infty$$ Thus, the limit for the given sequence does not exist.

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Most popular questions from this chapter

For a positive real number \(p,\) how do you interpret \(p^{p^{p \cdot *}},\) where the tower of exponents continues indefinitely? As it stands, the expression is ambiguous. The tower could be built from the top or from the bottom; that is, it could be evaluated by the recurrence relations \(a_{n+1}=p^{a_{n}}\) (building from the bottom) or \(a_{n+1}=a_{n}^{p}(\text { building from the top })\) where \(a_{0}=p\) in either case. The two recurrence relations have very different behaviors that depend on the value of \(p\). a. Use computations with various values of \(p > 0\) to find the values of \(p\) such that the sequence defined by (2) has a limit. Estimate the maximum value of \(p\) for which the sequence has a limit. b. Show that the sequence defined by (1) has a limit for certain values of \(p\). Make a table showing the approximate value of the tower for various values of \(p .\) Estimate the maximum value of \(p\) for which the sequence has a limit.

Determine whether the following series converge or diverge. $$\sum_{k=0}^{\infty} \frac{10}{k^{2}+9}$$

A well-known method for approximating \(\sqrt{c}\) for positive real numbers \(c\) consists of the following recurrence relation (based on Newton's method). Let \(a_{0}=c\) and $$a_{n+1}=\frac{1}{2}\left(a_{n}+\frac{c}{a_{n}}\right), \quad \text { for } n=0,1,2,3, \dots$$ a. Use this recurrence relation to approximate \(\sqrt{10} .\) How many terms of the sequence are needed to approximate \(\sqrt{10}\) with an error less than \(0.01 ?\) How many terms of the sequence are needed to approximate \(\sqrt{10}\) with an error less than \(0.0001 ?\) (To compute the error, assume a calculator gives the exact value.) b. Use this recurrence relation to approximate \(\sqrt{c},\) for \(c=2\) \(3, \ldots, 10 .\) Make a table showing how many terms of the sequence are needed to approximate \(\sqrt{c}\) with an error less than \(0.01.\)

Consider the following sequences defined by a recurrence relation. Use a calculator, analytical methods, and/or graphing to make a conjecture about the value of the limit or determine that the limit does not exist. $$a_{n+1}=\sqrt{2+a_{n}} ; a_{0}=1, n=0,1,2, \dots$$

Determine whether the following series converge absolutely or conditionally, or diverge. $$\sum_{k=1}^{\infty}\left(-\frac{1}{3}\right)^{k}$$
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