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Chapter 9: Problem 33

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

Short Answer

Expert verified
Question: Determine whether the limit of the sequence \(\left\\{ \frac{(-1)^{n} n}{n+1} \right\\}\) exists as \(n\) approaches infinity. Answer: The limit of the given sequence does not exist as \(n\) approaches infinity.

Step by step solution

01

Understand the sequence

The given sequence is: $$\left\\{\frac{(-1)^{n} n}{n+1}\right\\}$$ It is important to notice that it is an alternating sequence due to the \((-1)^n\) factor.
02

Find the limit of the absolute value of the sequence

To determine whether the sequence converges or diverges, we will find the limit of the absolute value of the sequence as \(n\) approaches infinity: $$\lim_{n \to \infty} \left| \frac{(-1)^{n} n}{n+1} \right|$$ Since \((-1)^n\) has no effect on the absolute value, we can disregard this term: $$\lim_{n \to \infty} \frac{n}{n+1}$$
03

Simplify the limit

To simplify the limit, we can use the highest power of \(n\) in the denominator to divide both the numerator and the denominator: $$\lim_{n \to \infty} \frac{\frac{n}{n}}{\frac{n+1}{n}}$$ $$\lim_{n \to \infty} \frac{1}{1 + \frac{1}{n}}$$
04

Evaluate the limit

Now, we can evaluate the limit as n approaches infinity: $$\lim_{n \to \infty} \frac{1}{1 + \frac{1}{n}} = \frac{1}{1+0} = 1$$ The limit of the absolute value of the sequence, \(|a_n|\), is equal to 1.
05

Determine if the limit of the sequence exist

Using the limit of the absolute value of the sequence, we can determine if the limit of the sequence exists. Since the limit of the absolute value is not zero, the limit of the given sequence does not exist: $$\lim_{n \to \infty} \frac{(-1)^{n} n}{n+1} \, \text{ does not exist}.$$ So, the limit of the given sequence does not exist.

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Most popular questions from this chapter

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

Determine whether the following series converge absolutely or conditionally, or diverge. $$\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k^{3 / 2}}$$

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. $$\sum_{k=1}^{\infty} \frac{x^{k}}{k}$$

Infinite products Use the ideas of Exercise 88 to evaluate the following infinite products. $$\text { a. } \prod_{k=0}^{\infty} e^{1 / 2^{k}}=e \cdot e^{1 / 2} \cdot e^{1 / 4} \cdot e^{1 / 8} \ldots$$ $$\text { b. } \prod_{k=2}^{\infty}\left(1-\frac{1}{k}\right)=\frac{1}{2} \cdot \frac{2}{3} \cdot \frac{3}{4} \cdot \frac{4}{5} \dots$$
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