1 Given that 1 Σ 1 z" with convergence in (-1, 1), find the power series for with center 3. Preview n=0 Identify its interval of convergence. The series is convergent from Preview , left end included (enter Y or N): , right end included (enter Y or N): to x = Preview
1 Given that 1 Σ 1 z" with convergence in (-1, 1), find the power series for with center 3. Preview n=0 Identify its interval of convergence. The series is convergent from Preview , left end included (enter Y or N): , right end included (enter Y or N): to x = Preview
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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![### Understanding Power Series and Convergence
#### Power Series Representation
Given the geometric series:
\[ \frac{1}{1 - x} = \sum_{n=0}^{\infty} x^n \]
which converges for \( x \) in the interval \((-1, 1)\), our goal is to find the power series for \(\frac{1}{x}\) with a center at 3.
#### Finding the Power Series
To find the power series for \(\frac{1}{x}\) centered at 3, start by manipulating the given geometric series.
```math
\sum_{n=0}^{\infty} \boxed{\phantom{This box is for entering the series}} \quad \text{Preview}
```
#### Interval of Convergence
Next, identify the interval of convergence of this series. The series is convergent in the same manner the original series converges within its specified interval, but now centered at 3.
Fill in the interval of convergence as follows:
\[
\text{from } x = \boxed{\text{enter lower bound}} \quad \text{Preview}
\]
\[
\text{to } x = \boxed{\text{enter upper bound}} \quad \text{Preview}
\]
**Indicate whether the endpoints are included by entering 'Y' for yes or 'N' for no:**
- Left end included: \(\boxed{\text{Y or N}}\)
- Right end included: \(\boxed{\text{Y or N}}\)
### Practical Application
Power series and their intervals of convergence are crucial in various applications within mathematics, including solving differential equations and evaluating integrals.
---
Feel free to input the required expressions and preview the series to visualize the results. Make sure to understand the convergence criteria and validate whether the endpoints are included in the interval.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F04df4d7d-8bb3-439f-93d2-ea8c38dfec4d%2F3ca5d94e-a054-410e-bdf4-17b611b5ded0%2F4c568d.png&w=3840&q=75)
Transcribed Image Text:### Understanding Power Series and Convergence
#### Power Series Representation
Given the geometric series:
\[ \frac{1}{1 - x} = \sum_{n=0}^{\infty} x^n \]
which converges for \( x \) in the interval \((-1, 1)\), our goal is to find the power series for \(\frac{1}{x}\) with a center at 3.
#### Finding the Power Series
To find the power series for \(\frac{1}{x}\) centered at 3, start by manipulating the given geometric series.
```math
\sum_{n=0}^{\infty} \boxed{\phantom{This box is for entering the series}} \quad \text{Preview}
```
#### Interval of Convergence
Next, identify the interval of convergence of this series. The series is convergent in the same manner the original series converges within its specified interval, but now centered at 3.
Fill in the interval of convergence as follows:
\[
\text{from } x = \boxed{\text{enter lower bound}} \quad \text{Preview}
\]
\[
\text{to } x = \boxed{\text{enter upper bound}} \quad \text{Preview}
\]
**Indicate whether the endpoints are included by entering 'Y' for yes or 'N' for no:**
- Left end included: \(\boxed{\text{Y or N}}\)
- Right end included: \(\boxed{\text{Y or N}}\)
### Practical Application
Power series and their intervals of convergence are crucial in various applications within mathematics, including solving differential equations and evaluating integrals.
---
Feel free to input the required expressions and preview the series to visualize the results. Make sure to understand the convergence criteria and validate whether the endpoints are included in the interval.
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