For the following function, find the Taylor series centered at x = 6 and give the first 5 nonzero terms of the Taylor series. Write the interval of convergence of the series. f(x) = f(x) = +... The interval of convergence is: + + n=1 f(x) + = ln(x) + (Give your answer in interval notation.) +
For the following function, find the Taylor series centered at x = 6 and give the first 5 nonzero terms of the Taylor series. Write the interval of convergence of the series. f(x) = f(x) = +... The interval of convergence is: + + n=1 f(x) + = ln(x) + (Give your answer in interval notation.) +
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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![### Problem Statement:
For the following function, find the Taylor series centered at \( x = 6 \) and give the first 5 nonzero terms of the Taylor series. Write the interval of convergence of the series.
#### Given Function:
\[ f(x) = \ln(x) \]
#### Taylor Series Representation:
\[ f(x) = \boxed{ } + \sum_{n=1}^{\infty} \boxed{ } \]
\[ f(x) = \boxed{ } + \boxed{ } + \boxed{ } + \boxed{ } + \boxed{ } + \cdots \]
#### Interval of Convergence:
The interval of convergence is: \[ \boxed{ } \] (Give your answer in interval notation.)
### Explanation:
In the text provided above, students are prompted to determine the Taylor series for the natural logarithm function \( f(x) = \ln(x) \) centered at \( x = 6 \). They are asked to provide the first five nonzero terms of this series and to determine the interval of convergence.
#### Steps to Solve:
1. **Derive the Taylor Series:**
- Use the formula for the Taylor series expansion of \( f(x) \) around a point \( a \):
\[ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n \]
- Compute the derivatives of \( \ln(x) \) and evaluate them at \( x = 6 \).
2. **Find the First Five Terms:**
- Substitute \( a = 6 \) and obtain the first five terms in the series expansion.
3. **Determine the Interval of Convergence:**
- Use the ratio test or other convergence tests to find the interval within which the series converges.
### Notes:
- The placeholders (\(\boxed{ }\)) are meant for students to fill in with their calculated terms and interval of convergence.
- Students should show their work to derive each term in the series and to determine the interval of convergence.
The above transcription can be included on an educational website to help students learn and practice finding Taylor series expansions and understanding intervals of convergence.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc5134d69-32d8-4659-bfc6-df6862919637%2F6e5dd043-f210-4ca0-bfed-380d7f2e7fc1%2Fk21yom7_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement:
For the following function, find the Taylor series centered at \( x = 6 \) and give the first 5 nonzero terms of the Taylor series. Write the interval of convergence of the series.
#### Given Function:
\[ f(x) = \ln(x) \]
#### Taylor Series Representation:
\[ f(x) = \boxed{ } + \sum_{n=1}^{\infty} \boxed{ } \]
\[ f(x) = \boxed{ } + \boxed{ } + \boxed{ } + \boxed{ } + \boxed{ } + \cdots \]
#### Interval of Convergence:
The interval of convergence is: \[ \boxed{ } \] (Give your answer in interval notation.)
### Explanation:
In the text provided above, students are prompted to determine the Taylor series for the natural logarithm function \( f(x) = \ln(x) \) centered at \( x = 6 \). They are asked to provide the first five nonzero terms of this series and to determine the interval of convergence.
#### Steps to Solve:
1. **Derive the Taylor Series:**
- Use the formula for the Taylor series expansion of \( f(x) \) around a point \( a \):
\[ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n \]
- Compute the derivatives of \( \ln(x) \) and evaluate them at \( x = 6 \).
2. **Find the First Five Terms:**
- Substitute \( a = 6 \) and obtain the first five terms in the series expansion.
3. **Determine the Interval of Convergence:**
- Use the ratio test or other convergence tests to find the interval within which the series converges.
### Notes:
- The placeholders (\(\boxed{ }\)) are meant for students to fill in with their calculated terms and interval of convergence.
- Students should show their work to derive each term in the series and to determine the interval of convergence.
The above transcription can be included on an educational website to help students learn and practice finding Taylor series expansions and understanding intervals of convergence.
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