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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can someone help me with this problem
![Certainly! Here is the transcription suitable for an educational website:
---
**Problem 7: Convergence Analysis**
Determine the radius of convergence and the interval of convergence for the following series, and justify your answer:
\[
\sum \frac{(-1)^k (x - 6)^k}{k^2 \cdot 10^k}
\]
---
**Explanation:**
To solve this problem, follow these steps:
1. **Identify the General Form:**
The given series is expressed in terms of powers of \((x - 6)\), indicating it is a power series centered at \(x = 6\).
2. **Apply the Ratio Test:**
The Ratio Test can be used to find the radius of convergence \(R\). For a series \(\sum a_k\), the Ratio Test considers the limit:
\[
L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right|
\]
If \(L < 1\), the series converges.
3. **Determine the Interval of Convergence:**
Once the radius \(R\) is found, the interval of convergence can be determined by examining endpoints and ensuring the series still converges when evaluated there.
Understanding these steps will allow you to correctly analyze the convergence behavior of the series.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbdd79cd3-887f-44e2-afdb-ed9bc8df797f%2Fa4263c42-efe5-432d-9f40-25485bffe2bc%2Fwomd8c9_processed.png&w=3840&q=75)
Transcribed Image Text:Certainly! Here is the transcription suitable for an educational website:
---
**Problem 7: Convergence Analysis**
Determine the radius of convergence and the interval of convergence for the following series, and justify your answer:
\[
\sum \frac{(-1)^k (x - 6)^k}{k^2 \cdot 10^k}
\]
---
**Explanation:**
To solve this problem, follow these steps:
1. **Identify the General Form:**
The given series is expressed in terms of powers of \((x - 6)\), indicating it is a power series centered at \(x = 6\).
2. **Apply the Ratio Test:**
The Ratio Test can be used to find the radius of convergence \(R\). For a series \(\sum a_k\), the Ratio Test considers the limit:
\[
L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right|
\]
If \(L < 1\), the series converges.
3. **Determine the Interval of Convergence:**
Once the radius \(R\) is found, the interval of convergence can be determined by examining endpoints and ensuring the series still converges when evaluated there.
Understanding these steps will allow you to correctly analyze the convergence behavior of the series.
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