/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 25 Find the limit of the following ... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 25

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

Short Answer

Expert verified
Answer: The limit of the sequence is $$\frac{1}{e}$$ as n approaches infinity.

Step by step solution

01

Recognize the sequence and its general term

In this case, the sequence is $$\left\\{\left(\frac{1}{n}\right)^{1 / n}\right\\}$$, and the general term (nth term) of the sequence can be written as: $$a_n=\left(\frac{1}{n}\right)^{1 / n}$$
02

Analyze the behavior of the sequence as n approaches infinity

To find the limit of the given sequence, we need to find the value of the general term as n approaches infinity, or formally: $$\lim_{n\to\infty}\left(\frac{1}{n}\right)^{1/n}$$
03

Apply a known limit

To solve this limit, we can apply a known limit called the "Exponential Limit": $$\lim_{n\to\infty}\left(1+\frac{a}{n}\right)^n=e^a$$, where a is a constant. We can rewrite our original limit to resemble this known limit by making the following substitutions: let $$a=-n$$ and $$b=\frac{1}{n}$$. Therefore, our new limit becomes $$\lim_{n\to\infty}\left(1-\frac{1}{n}\right)^{-n}$$ Now, we can directly apply the Exponential Limit, which gives us: $$\lim_{n\to\infty}\left(1-\frac{1}{n}\right)^{-n}=e^{-1}=\frac{1}{e}$$
04

State the result

As a result, the limit of the given sequence is the value of the general term as n approaches infinity: $$\lim_{n\to\infty}\left(\frac{1}{n}\right)^{1/n}=\frac{1}{e}$$

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Most popular questions from this chapter

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. $$a_{n+1}=\frac{1}{2} a_{n}+2 ; a_{0}=5, n=0,1,2, \dots$$

The Riemann zeta function is the subject of extensive research and is associated with several renowned unsolved problems. It is defined by \(\zeta(x)=\sum_{k=1}^{\infty} \frac{1}{k^{x}}\). When \(x\) is a real number, the zeta function becomes a \(p\) -series. For even positive integers \(p,\) the value of \(\zeta(p)\) is known exactly. For example, $$ \sum_{k=1}^{\infty} \frac{1}{k^{2}}=\frac{\pi^{2}}{6}, \quad \sum_{k=1}^{\infty} \frac{1}{k^{4}}=\frac{\pi^{4}}{90}, \quad \text { and } \quad \sum_{k=1}^{\infty} \frac{1}{k^{6}}=\frac{\pi^{6}}{945}, \ldots $$ Use estimation techniques to approximate \(\zeta(3)\) and \(\zeta(5)\) (whose values are not known exactly) with a remainder less than \(10^{-3}\).

Consider the following situations that generate a sequence. a. Write out the first five terms of the sequence. b. Find an explicit formula for the terms of the sequence. c. Find a recurrence relation that generates the sequence. d. Using a calculator or a graphing utility, estimate the limit of the sequence or state that it does not exist. The Consumer Price Index (the CPI is a measure of the U.S. cost of living) is given a base value of 100 in the year \(1984 .\) Assume the CPI has increased by an average of \(3 \%\) per year since \(1984 .\) Let \(c_{n}\) be the CPI \(n\) years after \(1984,\) where \(c_{0}=100.\)

Suppose a ball is thrown upward to a height of \(h_{0}\) meters. Each time the ball bounces, it 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 \(\left\\{S_{n}\right\\}.\) $$h_{0}=20, r=0.75$$

A glimpse ahead to power series Use the Ratio Test to determine the values of \(x \geq 0\) for which each series converges. $$\sum_{k=1}^{\infty} \frac{x^{2 k}}{k^{2}}$$
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