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Chapter 9: Problem 2

Give an example of a nondecreasing sequence without a limit.

Short Answer

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Question: Provide an example of a nondecreasing sequence without a limit and explain why it doesn't have a limit. Answer: An example of a nondecreasing sequence without a limit is the sequence defined by the function \(a_n = n\), with the terms {1, 2, 3, 4, ...}. This sequence doesn't have a limit because for any real number L, there exists a term a_n in the sequence such that a_n > L, meaning the sequence approaches infinity and does not converge to a finite value.

Step by step solution

01

Define the nondecreasing sequence

Let us consider a simple nondecreasing sequence, where the nth term of the sequence (a_n) is given by the function: \[ a_n = n \] This sequence is nondecreasing because the terms increase as we move forward in the sequence.
02

Show that the sequence does not have a limit

To show that the sequence does not have a limit, we have to prove that for any real number L, there exists a term a_n in the sequence such that a_n > L. Let's consider an arbitrary real number L. Since there are infinitely many integers, we can find an integer N such that N > L. For that N, the term a_N in the sequence becomes: \[ a_N = N \] Since N > L, we get, \[ a_N > L \] This inequality shows that no matter how large L is, there's always a term a_N in the sequence that is greater than L. Thus, this nondecreasing sequence does not have a limit.
03

Provide the example

The sequence we have found is a valid example of a nondecreasing sequence without a limit. The sequence is given by: \[ a_n = n \] with the terms {1, 2, 3, 4, ...}. This sequence does not have a finite limit, as it approaches infinity.

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Most popular questions from this chapter

Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than \(10^{-4} .\) Although you do not need it, the exact value of the series is given in each case. $$\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{(2 k+1) !}$$

Determine whether the following statements are true and give an explanation or counterexample. a. A series that converges must converge absolutely. b. A series that converges absolutely must converge. c. A series that converges conditionally must converge. d. If \(\sum a_{k}\) diverges, then \(\Sigma\left|a_{k}\right|\) diverges. e. If \(\sum a_{k}^{2}\) converges, then \(\sum a_{k}\) converges. f. If \(a_{k}>0\) and \(\sum a_{k}\) converges, then \(\Sigma a_{k}^{2}\) converges. g. If \(\Sigma a_{k}\) converges conditionally, then \(\Sigma\left|a_{k}\right|\) diverges.

Consider the following infinite series. a. Write out the first four terms of the sequence of partial sums. b. Estimate the limit of \(\left\\{S_{n}\right\\}\) or state that it does not exist. $$\sum_{k=1}^{\infty} 3^{-k}$$

Consider the sequence \(\left\\{F_{n}\right\\}\) defined by $$F_{n}=\sum_{k=1}^{\infty} \frac{1}{k(k+n)^{\prime}}$$ for \(n=0,1,2, \ldots . .\) When \(n=0,\) the series is a \(p\) -series, and we have \(F_{0}=\pi^{2} / 6\) (Exercises 65 and 66 ). for \(n=0,1,2, \ldots . .\) When \(n=0,\) the series is a \(p\) -series, and we have \(F_{0}=\pi^{2} / 6\) (Exercises 65 and 66 ).

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. Jack took a \(200-\mathrm{mg}\) dose of a strong painkiller at midnight. Every hour, \(5 \%\) of the drug is washed out of his bloodstream. Let \(d_{n}\) be the amount of drug in Jack's blood \(n\) hours after the drug was taken, where \(d_{0}=200 \mathrm{mg}.\)
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