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Chapter 8: Problem 1

Give an example of a nonincreasing sequence with a limit.

Short Answer

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Question: Give an example of a nonincreasing sequence that converges to a limit and determine its limit. Answer: An example of a nonincreasing sequence that converges to a limit is the sequence (a_n) = 1/n for n 鈮� 1. The sequence looks like: 1/1, 1/2, 1/3, .... The limit of this sequence is 0 as n approaches infinity.

Step by step solution

01

Example of a nonincreasing sequence

Let's consider the sequence \((a_n)\) defined by \(a_n = \frac{1}{n}\) for \(n \geq 1\). This sequence is nonincreasing since each term is smaller than or equal to the previous term, as \(n\) increases. The sequence will look like this: \(\frac{1}{1}, \frac{1}{2}, \frac{1}{3}, \ldots\)
02

Determine the limit

To find the limit, we need to examine what happens to the sequence as \(n\) approaches infinity. We can do this by finding \(\lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{1}{n}\).
03

Use properties of limits

We know that \(\lim_{n \to \infty} \frac{1}{n} = 0\). This is because as \(n\) increases without bound, the fraction \(\frac{1}{n}\) gets smaller and smaller, approaching zero.
04

Conclusion

We found that the nonincreasing sequence \((a_n) = \frac{1}{n}\) has a limit, which is 0. Therefore, the example of a nonincreasing sequence with a limit is: \(\frac{1}{1}, \frac{1}{2}, \frac{1}{3}, \ldots\) with limit 0.

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Most popular questions from this chapter

Consider the sequence \(\left\\{x_{n}\right\\}\) defined for \(n=1,2,3, \ldots\) by $$x_{n}=\sum_{k=n+1}^{2 n} \frac{1}{k}=\frac{1}{n+1}+\frac{1}{n+2}+\dots+\frac{1}{2 n}.$$ a. Write out the terms \(x_{1}, x_{2}, x_{3}\). b. Show that \(\frac{1}{2} \leq x_{n}<1,\) for \(n=1,2,3, \ldots\). c. Show that \(x_{n}\) is the right Riemann sum for \(\int_{1}^{2} \frac{d x}{x}\) using \(n\) subintervals. d. Conclude that \(\lim _{n \rightarrow \infty} x_{n}=\ln 2\).

A heifer weighing 200 lb today gains 5 lb per day with a food cost of \(45 \mathrm{c} /\) day. The price for heifers is \(65 \mathrm{q} / \mathrm{lb}\) today but is falling \(1 \% /\) day. a. Let \(h_{n}\) be the profit in selling the heifer on the \(n\) th day, where \(h_{0}=(200 \mathrm{lb}) \cdot(\$ 0.65 / \mathrm{lb})=\$ 130 .\) Write out the first 10 terms of the sequence \(\left\\{h_{n}\right\\}\). b. How many days after today should the heifer be sold to maximize the profit?

Here is a fascinating (unsolved) problem known as the hailstone problem (or the Ulam Conjecture or the Collatz Conjecture). It involves sequences in two different ways. First, choose a positive integer \(N\) and call it \(a_{0} .\) This is the seed of a sequence. The rest of the sequence is generated as follows: For \(n=0,1,2, \ldots\) $$a_{n+1}=\left\\{\begin{array}{ll} a_{n} / 2 & \text { if } a_{n} \text { is even } \\ 3 a_{n}+1 & \text { if } a_{n} \text { is odd .} \end{array}\right.$$ However, if \(a_{n}=1\) for any \(n,\) then the sequence terminates. a. Compute the sequence that results from the seeds \(N=2,3\), \(4, \ldots, 10 .\) You should verify that in all these cases, the sequence eventually terminates. The hailstone conjecture (still unproved) states that for all positive integers \(N\), the sequence terminates after a finite number of terms. b. Now define the hailstone sequence \(\left\\{H_{k}\right\\},\) which is the number of terms needed for the sequence \(\left\\{a_{n}\right\\}\) to terminate starting with a seed of \(k\). Verify that \(H_{2}=1, H_{3}=7\), and \(H_{4}=2\). c. Plot as many terms of the hailstone sequence as is feasible. How did the sequence get its name? Does the conjecture appear to be true?

Use Theorem 8.6 to find the limit of the following sequences or state that they diverge. $$\left\\{\frac{n^{10}}{\ln ^{1000} n}\right\\}$$

Consider the alternating series $$ \sum_{k=1}^{\infty}(-1)^{k+1} a_{k}, \text { where } a_{k}=\left\\{\begin{array}{cl} \frac{4}{k+1}, & \text { if } k \text { is odd } \\ \frac{2}{k}, & \text { if } k \text { is even } \end{array}\right. $$ a. Write out the first ten terms of the series, group them in pairs, and show that the even partial sums of the series form the (divergent) harmonic series. b. Show that \(\lim _{k \rightarrow \infty} a_{k}=0\) c. Explain why the series diverges even though the terms of the series approach zero.
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