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Chapter 8: Problem 2

Give an example of a nondecreasing sequence without a limit.

Short Answer

Expert verified
Question: Create a non-decreasing sequence without a limit and verify its properties. Answer: The sequence a_n = n, where n 鈭� 鈩� (the natural numbers) is a non-decreasing sequence without a limit. This sequence satisfies the non-decreasing property as each term is greater than the previous term. Additionally, the sequence grows indefinitely and does not approach a specific value, indicating that it does not have a limit.

Step by step solution

01

Choose a Non-decreasing Pattern

We need to create a sequence where each term is either the same or greater than the previous term. One of the simplest examples is to choose a pattern where each term is the next natural number in line. Let's use the sequence of natural numbers as our base.
02

Create a Sequence Without a Limit

Now that we have a non-decreasing pattern, we must ensure that it doesn't have a limit. A sequence of natural numbers grows indefinitely, never reaching a limit. Thus, our sequence satisfies the requirement of not having a limit. The sequence is as follows: a_n = n, where n 鈭� 鈩�.
03

Verify the Sequence

Now that we have created a sequence, let's verify that it is non-decreasing and without a limit. The sequence is a_n = n, where n 鈭� 鈩�. As we progress through the sequence, we move through the natural numbers {1, 2, 3, 4, ...}. As we progress, each term is greater than the previous term, satisfying the non-decreasing requirement. Additionally, the sequence grows indefinitely with no specific value it approaches, so it does not have a limit.
04

Conclusion

We have created a non-decreasing sequence without a limit. The sequence is the natural numbers, a_n = n, where n 鈭� 鈩�. This sequence demonstrates the requirement of being non-decreasing and not having a limit, as each term is greater than the previous term and the sequence grows indefinitely.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sequence Limit
In mathematics, a sequence limit refers to the value that the terms of a sequence approach as the index becomes indefinitely large. Understanding a sequence limit is crucial when analyzing the behavior of sequences. A sequence \(\{a_n\}\) is said to have a limit if, as \(n\) approaches infinity, the terms of the sequence get arbitrarily close to a certain number L. This means:
  • For every positive number \(\epsilon\), there exists an integer N such that for all \(n > N\), the absolute difference between \(a_n\) and L is less than \(\epsilon\).
  • If such an L exists, it is the limit of the sequence.

However, not all sequences have a limit. For example, the sequence of natural numbers \(\{1, 2, 3, ...\}\) grows indefinitely. It does not approach a finite value as \(n\) increases, thus it does not possess a sequence limit.
Natural Numbers
Natural numbers are a fundamental concept in mathematics and form the basic building block for arithmetic. They are positive integers starting from 1 and increasing indefinitely: \(\{1, 2, 3, 4, ...\}\).
  • Natural numbers are used for counting and ordering.
  • They continue infinitely, meaning there is no largest natural number.

In the context of sequences, using natural numbers is a common practice, particularly because they form a simple, regular sequence that is easy to work with. Sequences based on natural numbers, like \(a_n = n\), result in a straightforward progression that is both non-decreasing and without a limit. This makes them an ideal example when discussing concepts such as non-decreasing sequences and sequences without a limit.
Mathematical Sequences
A mathematical sequence is a set of numbers in a specific order. Each number in the sequence is called a term. Mathematical sequences can either be finite, with a specific number of terms, or infinite, continuing indefinitely.
  • There are many different types of sequences, such as arithmetic (where the difference between terms is constant) and geometric (where the ratio between terms is constant).
  • In this case, a non-decreasing sequence is one where each subsequent term is greater than or equal to the previous one.

An important aspect of sequences is their behavior as the number of terms increases. Some sequences may converge towards a limit, while others, like the natural numbers, do not. Understanding the characteristics of sequences, such as whether they are increasing, decreasing, or cyclical, helps in analyzing their behavior and properties.

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Most popular questions from this chapter

Evaluate the limit of the following sequences or state that the limit does not exist. $$a_{n}=\frac{7^{n}}{n^{7} 5^{n}}$$

A fishery manager knows that her fish population naturally increases at a rate of \(1.5 \%\) per month, while 80 fish are harvested each month. Let \(F_{n}\) be the fish population after the \(n\) th month, where \(F_{0}=4000\) fish. a. Write out the first five terms of the sequence \(\left\\{F_{n}\right\\}\). b. Find a recurrence relation that generates the sequence \(\left\\{F_{n}\right\\}\). c. Does the fish population decrease or increase in the long run? d. Determine whether the fish population decreases or increases in the long run if the initial population is 5500 fish. e. Determine the initial fish population \(F_{0}\) below which the population decreases.

Use the formal definition of the limit of a sequence to prove the following limits. $$\lim _{n \rightarrow \infty} \frac{1}{n}=0$$

\(\sum_{k=1}^{\infty} \frac{1}{k^{2}}=\frac{\pi^{2}}{6} \operatorname{In} 1734,\) Leonhard Euler informally proved that \(\sum_{k=1}^{\infty} \frac{1}{k^{2}}=\frac{\pi^{2}}{6} .\) An elegant proof is outlined here that uses the inequality $$\cot ^{2} x<\frac{1}{x^{2}}<1+\cot ^{2} x\left(\text { provided that } 0

An infinite product \(P=a_{1} a_{2} a_{3} \ldots,\) which is denoted \(\prod_{k=1}^{\infty} a_{k}\) is the limit of the sequence of partial products \(\left\\{a_{1}, a_{1} a_{2}, a_{1} a_{2} a_{3}, \ldots\right\\} .\) Assume that \(a_{k}>0\) for all \(k\) a. Show that the infinite product converges (which means its sequence of partial products converges) provided the series \(\sum_{k=1}^{\infty} \ln a_{k}\) converges. b. Consider the infinite product $$P=\prod_{k=2}^{\infty}\left(1-\frac{1}{k^{2}}\right)=\frac{3}{4} \cdot \frac{8}{9} \cdot \frac{15}{16} \cdot \frac{24}{25} \cdots$$ Write out the first few terms of the sequence of partial products, $$P_{n}=\prod_{k=2}^{n}\left(1-\frac{1}{k^{2}}\right)$$ (for example, \(P_{2}=\frac{3}{4}, P_{3}=\frac{2}{3}\) ). Write out enough terms to determine the value of \(P=\lim _{n \rightarrow \infty} P_{n}\) c. Use the results of parts (a) and (b) to evaluate the series $$\sum_{k=2}^{\infty} \ln \left(1-\frac{1}{k^{2}}\right)$$
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