and (b) use a graphing utility to graph the first 10 terms of the to show that the sequence with the given nth term converges, In Exercises 61-64, (a) use Theorem 9.5 1 62. a, = 5 - 2 %3D 61. a, = 7 + %3D n
and (b) use a graphing utility to graph the first 10 terms of the to show that the sequence with the given nth term converges, In Exercises 61-64, (a) use Theorem 9.5 1 62. a, = 5 - 2 %3D 61. a, = 7 + %3D n
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Sorry for duplicate question. forgot to put the number I need answered. it is #62.
![**Transcription for Educational Website**
---
### Sequences and Limit Theorems
**Exercises 61-64**
*Using a Theorem:* Utilize Theorem 9.5 to demonstrate the convergence of the sequence with the given nth term. Additionally, use a graphing utility to plot the first 10 terms of the sequence and determine its limit.
- **Exercise 61:** \( a_n = 7 + \frac{1}{n} \)
- **Exercise 62:** \( a_n = 5 - \frac{2}{n} \)
- **Exercise 63:** \( a_n = \frac{1}{3} \left( 1 - \frac{1}{3^n} \right) \)
- **Exercise 64:** \( a_n = 2 + \frac{1}{5^n} \)
**Conceptual Discussions**
**65. Increasing Sequence**
Let \(\{ a_n \} \) be an increasing sequence such that \(2 \leq a_n \leq 4\). Explain why \(\{ a_n \} \) has a limit. What can you conclude about the limit?
**66. Monotonic Sequence**
Let \(\{ a_n \} \) be a monotonic sequence such that \(a_n \leq 1\). Discuss the convergence of \(\{ a_n \} \). When \(\{ a_n \} \) converges, what can you conclude about its limit?
**67. Compound Interest**
Consider the sequence \(\{ A_n \} \) whose nth term is given by:
\[ A_n = P \left(1 + \frac{r}{12}\right)^n \]
*Note: There is a background image of a stack of dollar bills to visually represent the concept of compound interest.*
---
In this educational excerpt, students are encouraged to explore the concepts of convergence and limits in sequences through practical exercises and real-world applications like compound interest.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7da000f5-964a-4988-803f-c8092a11ccf6%2Fe47215d5-f887-4910-8ded-8f06ab622593%2Fchdoet_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Transcription for Educational Website**
---
### Sequences and Limit Theorems
**Exercises 61-64**
*Using a Theorem:* Utilize Theorem 9.5 to demonstrate the convergence of the sequence with the given nth term. Additionally, use a graphing utility to plot the first 10 terms of the sequence and determine its limit.
- **Exercise 61:** \( a_n = 7 + \frac{1}{n} \)
- **Exercise 62:** \( a_n = 5 - \frac{2}{n} \)
- **Exercise 63:** \( a_n = \frac{1}{3} \left( 1 - \frac{1}{3^n} \right) \)
- **Exercise 64:** \( a_n = 2 + \frac{1}{5^n} \)
**Conceptual Discussions**
**65. Increasing Sequence**
Let \(\{ a_n \} \) be an increasing sequence such that \(2 \leq a_n \leq 4\). Explain why \(\{ a_n \} \) has a limit. What can you conclude about the limit?
**66. Monotonic Sequence**
Let \(\{ a_n \} \) be a monotonic sequence such that \(a_n \leq 1\). Discuss the convergence of \(\{ a_n \} \). When \(\{ a_n \} \) converges, what can you conclude about its limit?
**67. Compound Interest**
Consider the sequence \(\{ A_n \} \) whose nth term is given by:
\[ A_n = P \left(1 + \frac{r}{12}\right)^n \]
*Note: There is a background image of a stack of dollar bills to visually represent the concept of compound interest.*
---
In this educational excerpt, students are encouraged to explore the concepts of convergence and limits in sequences through practical exercises and real-world applications like compound interest.
![**Bounded Sequences**
1. A sequence \(\{a_n\}\) is **bounded below** when there is a real number \(N\) such that \(N \leq a_n\) for all \(n\). The number \(N\) is called a **lower bound** of the sequence.
2. A sequence \(\{a_n\}\) is **bounded above** when there is a real number \(M\) such that \(a_n \leq M\) for all \(n\). The number \(M\) is called an **upper bound** of the sequence.
3. A sequence \(\{a_n\}\) is **bounded** when it is bounded both above and below.
To see this, note that:
\[ 2 \leq a_n \leq 4, \quad 1 \leq b_n \leq 2, \quad \text{and} \quad 0 \leq c_n \leq \frac{4}{3} \]
One important property of the real numbers is that they are **complete**. Informally, this means there are no holes or gaps on the real number line. (The set of rational numbers does not have the completeness property.) The completeness axiom of real numbers states that if a sequence has an upper bound, then it must have a **least upper bound**. For example, the least upper bound of the sequence \(\{a_n\} = \{n/(n+1)\}\) is:
\[
\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \ldots, \frac{n}{n+1}
\]
**Theorem 9.5: Bounded Monotonic Sequences**
If a sequence \(\{a_n\}\) is bounded and monotonic, then it **converges**.
**Proof**
Assume that the sequence is nondecreasing. For simplicity, assume that each term in the sequence is positive. If the sequence is bounded, there must exist an upper bound \(M\) such that:
\[ a_1 \leq a_2 \leq a_3 \leq \cdots \leq a_n \leq \cdots \leq M \]
From the completeness axiom, it follows that there](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7da000f5-964a-4988-803f-c8092a11ccf6%2Fe47215d5-f887-4910-8ded-8f06ab622593%2F8nhcovo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Bounded Sequences**
1. A sequence \(\{a_n\}\) is **bounded below** when there is a real number \(N\) such that \(N \leq a_n\) for all \(n\). The number \(N\) is called a **lower bound** of the sequence.
2. A sequence \(\{a_n\}\) is **bounded above** when there is a real number \(M\) such that \(a_n \leq M\) for all \(n\). The number \(M\) is called an **upper bound** of the sequence.
3. A sequence \(\{a_n\}\) is **bounded** when it is bounded both above and below.
To see this, note that:
\[ 2 \leq a_n \leq 4, \quad 1 \leq b_n \leq 2, \quad \text{and} \quad 0 \leq c_n \leq \frac{4}{3} \]
One important property of the real numbers is that they are **complete**. Informally, this means there are no holes or gaps on the real number line. (The set of rational numbers does not have the completeness property.) The completeness axiom of real numbers states that if a sequence has an upper bound, then it must have a **least upper bound**. For example, the least upper bound of the sequence \(\{a_n\} = \{n/(n+1)\}\) is:
\[
\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \ldots, \frac{n}{n+1}
\]
**Theorem 9.5: Bounded Monotonic Sequences**
If a sequence \(\{a_n\}\) is bounded and monotonic, then it **converges**.
**Proof**
Assume that the sequence is nondecreasing. For simplicity, assume that each term in the sequence is positive. If the sequence is bounded, there must exist an upper bound \(M\) such that:
\[ a_1 \leq a_2 \leq a_3 \leq \cdots \leq a_n \leq \cdots \leq M \]
From the completeness axiom, it follows that there
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