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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![**Sequence Pattern Analysis and Formula Derivation**
To determine the formula for the general term \( a_n \) of the given sequence, let's first examine the sequence itself:
\[
\left\{ - \frac{12}{3}, \frac{14}{9}, - \frac{16}{27}, \frac{18}{81}, - \frac{20}{243}, \cdots \right\}
\]
**Step-by-Step Analysis:**
1. **Numerator Analysis:**
- The numerators are: \( 12, 14, 16, 18, 20, \ldots \).
- Observation: These numbers form an arithmetic sequence with a common difference of \( 2 \).
- Therefore, the \( n \)-th term of the numerator sequence can be written as:
\[ N = 12 + (n-1) \times 2 = 2n + 10 \]
2. **Denominator Analysis:**
- The denominators are: \( 3, 9, 27, 81, 243, \ldots \).
- Observation: These numbers form a geometric sequence with a common ratio of \( 3 \).
- Therefore, the \( n \)-th term of the denominator sequence can be written as:
\[ D = 3^n \]
3. **Sign Analysis:**
- The terms alternatively switch between negative and positive.
- Therefore, the sign can be represented using \( (-1)^n \) since \( (-1)^1 = -1 \) (first term is negative), \( (-1)^2 = 1 \) (second term is positive), and so on.
**Combining These Observations:**
Putting these observations together, the general term \( a_n \) of the sequence is given by:
\[ a_n = (-1)^n \times \frac{2n + 10}{3^n} \]
**Final Equation:**
\[
a_n = (-1)^n \times \frac{2n + 10}{3^n}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc31d976c-acd4-451e-8eb9-fde305025b60%2Fc59f7a97-15b8-4e2b-83ee-60ce7c868a54%2F8i6eh3u_processed.png&w=3840&q=75)
Transcribed Image Text:**Sequence Pattern Analysis and Formula Derivation**
To determine the formula for the general term \( a_n \) of the given sequence, let's first examine the sequence itself:
\[
\left\{ - \frac{12}{3}, \frac{14}{9}, - \frac{16}{27}, \frac{18}{81}, - \frac{20}{243}, \cdots \right\}
\]
**Step-by-Step Analysis:**
1. **Numerator Analysis:**
- The numerators are: \( 12, 14, 16, 18, 20, \ldots \).
- Observation: These numbers form an arithmetic sequence with a common difference of \( 2 \).
- Therefore, the \( n \)-th term of the numerator sequence can be written as:
\[ N = 12 + (n-1) \times 2 = 2n + 10 \]
2. **Denominator Analysis:**
- The denominators are: \( 3, 9, 27, 81, 243, \ldots \).
- Observation: These numbers form a geometric sequence with a common ratio of \( 3 \).
- Therefore, the \( n \)-th term of the denominator sequence can be written as:
\[ D = 3^n \]
3. **Sign Analysis:**
- The terms alternatively switch between negative and positive.
- Therefore, the sign can be represented using \( (-1)^n \) since \( (-1)^1 = -1 \) (first term is negative), \( (-1)^2 = 1 \) (second term is positive), and so on.
**Combining These Observations:**
Putting these observations together, the general term \( a_n \) of the sequence is given by:
\[ a_n = (-1)^n \times \frac{2n + 10}{3^n} \]
**Final Equation:**
\[
a_n = (-1)^n \times \frac{2n + 10}{3^n}
\]
![## Telescoping Series Example
Consider the series:
\[ \ln\left(\frac{n}{n+1}\right) \]
Express this series as a telescoping series:
\[ \sum_{n=1}^{\infty} \ln\left(\frac{n}{n+1}\right) = \sum_{n=1}^{\infty} \]
Evaluate this series or enter DNE if the series diverges:
\[ S = \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc31d976c-acd4-451e-8eb9-fde305025b60%2Fc59f7a97-15b8-4e2b-83ee-60ce7c868a54%2Fxzacli4_processed.png&w=3840&q=75)
Transcribed Image Text:## Telescoping Series Example
Consider the series:
\[ \ln\left(\frac{n}{n+1}\right) \]
Express this series as a telescoping series:
\[ \sum_{n=1}^{\infty} \ln\left(\frac{n}{n+1}\right) = \sum_{n=1}^{\infty} \]
Evaluate this series or enter DNE if the series diverges:
\[ S = \]
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