1 Given that Σ 1 x" with convergence in (-1, 1), find the power series for with center 1- x 1- 3x9 n=0 0. 00
1 Given that Σ 1 x" with convergence in (-1, 1), find the power series for with center 1- x 1- 3x9 n=0 0. 00
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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![**Topic: Power Series and Interval of Convergence**
**Given:**
- The geometric series identity:
\[
\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n
\]
with convergence in the interval \((-1, 1)\).
**Task:**
- Find the power series for
\[
\frac{1}{1-3x^9}
\]
with the center at 0.
**Solution:**
To find the power series expression:
\[
\sum_{n=0}^{\infty} (3x^9)^n
\]
**Identify the Interval of Convergence:**
The series is convergent from:
- \(x = \underline{\phantom{0}}\), left end included (enter Y or N): \(\underline{\phantom{Y/N}}\)
- to \(x = \underline{\phantom{0}}\), right end included (enter Y or N): \(\underline{\phantom{Y/N}}\)
**Explanation:**
Based on the geometric series formula, we substitute \(3x^9\) for \(x\). This substitution affects the interval of convergence, requiring the expression \(|3x^9| < 1\) to determine endpoints.
Complete the interval a evaluations and edge inclusion decisions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F62b092f3-63db-4d93-982f-67d0473d8e68%2Fa7413386-144b-4d80-b3c5-588cf202fdaa%2Fvnj9sc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Topic: Power Series and Interval of Convergence**
**Given:**
- The geometric series identity:
\[
\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n
\]
with convergence in the interval \((-1, 1)\).
**Task:**
- Find the power series for
\[
\frac{1}{1-3x^9}
\]
with the center at 0.
**Solution:**
To find the power series expression:
\[
\sum_{n=0}^{\infty} (3x^9)^n
\]
**Identify the Interval of Convergence:**
The series is convergent from:
- \(x = \underline{\phantom{0}}\), left end included (enter Y or N): \(\underline{\phantom{Y/N}}\)
- to \(x = \underline{\phantom{0}}\), right end included (enter Y or N): \(\underline{\phantom{Y/N}}\)
**Explanation:**
Based on the geometric series formula, we substitute \(3x^9\) for \(x\). This substitution affects the interval of convergence, requiring the expression \(|3x^9| < 1\) to determine endpoints.
Complete the interval a evaluations and edge inclusion decisions.
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