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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![### Determining the Convergence of a Series
Find all the values of \( x \) such that the given series would converge:
\[
\sum_{n=1}^{\infty} \frac{(-1)^n x^n}{6^n (n^2 + 9)}
\]
The series is convergent within the interval:
from \( x = \) \_\_\_ , left end included (enter Y or N): \_\_
to \( x = \) \_\_\_ , right end included (enter Y or N): \_\_
### Explanation
The given series is:
\[
\sum_{n=1}^{\infty} \frac{(-1)^n x^n}{6^n (n^2 + 9)}
\]
To determine the range of \( x \) for which this series converges, one will typically use the ratio test or the root test to find the radius of convergence. Once the radius of convergence \( R \) is found, the interval of convergence will be \(( -R, R )\). You also need to check the endpoints \(-R\) and \(R\) separately to determine whether the series converges at those points or not.
#### Steps to solve the problem:
1. **Apply the Ratio Test or Root Test:** These tests help in determining how fast the terms of the series approach zero.
2. **Find the Radius of Convergence \( R \):** This is determined from the outcome of the Ratio or Root Test.
3. **Check Endpoint Convergence:** Substitute \( x = -R \) and \( x = R \) into the series to see if it converges at those values.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F17004842-8eab-4d8d-951d-53ad221d4829%2F50f7715d-c1c2-421e-9632-45f2c659e349%2Fqqlodhb.jpeg&w=3840&q=75)
Transcribed Image Text:### Determining the Convergence of a Series
Find all the values of \( x \) such that the given series would converge:
\[
\sum_{n=1}^{\infty} \frac{(-1)^n x^n}{6^n (n^2 + 9)}
\]
The series is convergent within the interval:
from \( x = \) \_\_\_ , left end included (enter Y or N): \_\_
to \( x = \) \_\_\_ , right end included (enter Y or N): \_\_
### Explanation
The given series is:
\[
\sum_{n=1}^{\infty} \frac{(-1)^n x^n}{6^n (n^2 + 9)}
\]
To determine the range of \( x \) for which this series converges, one will typically use the ratio test or the root test to find the radius of convergence. Once the radius of convergence \( R \) is found, the interval of convergence will be \(( -R, R )\). You also need to check the endpoints \(-R\) and \(R\) separately to determine whether the series converges at those points or not.
#### Steps to solve the problem:
1. **Apply the Ratio Test or Root Test:** These tests help in determining how fast the terms of the series approach zero.
2. **Find the Radius of Convergence \( R \):** This is determined from the outcome of the Ratio or Root Test.
3. **Check Endpoint Convergence:** Substitute \( x = -R \) and \( x = R \) into the series to see if it converges at those values.
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