Find the radius of convergence of the power series. (-1)"x" Σn=08"
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![**Problem Statement:**
Find the radius of convergence of the power series.
**Mathematical Expression:**
\[
\sum_{n=0}^{\infty} \frac{(-1)^n x^n}{9^n}
\]
**Explanation:**
The given power series is:
\[
\sum_{n=0}^{\infty} \frac{(-1)^n x^n}{9^n} = \sum_{n=0}^{\infty} \left(\frac{(-1)x}{9}\right)^n
\]
In this series, each term is of the form:
\[
a_n = \left(\frac{(-1)x}{9}\right)^n
\]
This is similar in form to a geometric series \(\sum_{n=0}^{\infty} r^n\), where \(r = \frac{(-1)x}{9}\).
The radius of convergence \(R\) is determined by the condition \(|r| < 1\).
Thus, for convergence:
\[
\left|\frac{x}{9}\right| < 1
\]
\[
|x| < 9
\]
**Conclusion:**
The radius of convergence of the series is \(R = 9\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F45235a53-69d6-46fa-83a8-de0bf8bf66d1%2F7b163338-49f4-45e8-8b32-58f4211003b4%2Fl7zhsee_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the radius of convergence of the power series.
**Mathematical Expression:**
\[
\sum_{n=0}^{\infty} \frac{(-1)^n x^n}{9^n}
\]
**Explanation:**
The given power series is:
\[
\sum_{n=0}^{\infty} \frac{(-1)^n x^n}{9^n} = \sum_{n=0}^{\infty} \left(\frac{(-1)x}{9}\right)^n
\]
In this series, each term is of the form:
\[
a_n = \left(\frac{(-1)x}{9}\right)^n
\]
This is similar in form to a geometric series \(\sum_{n=0}^{\infty} r^n\), where \(r = \frac{(-1)x}{9}\).
The radius of convergence \(R\) is determined by the condition \(|r| < 1\).
Thus, for convergence:
\[
\left|\frac{x}{9}\right| < 1
\]
\[
|x| < 9
\]
**Conclusion:**
The radius of convergence of the series is \(R = 9\).
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