Find the radius of convergence for 2.4.6... (2n) (2n)! n=1 ·xn.
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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6.1.6
![**Find the Radius of Convergence**
Consider the series:
\[
\sum_{n=1}^{\infty} \frac{2 \cdot 4 \cdot 6 \cdot \ldots \cdot (2n)}{(2n)!} x^n
\]
To determine the radius of convergence for this series, analyze the pattern and apply mathematical techniques such as the ratio test or root test. In this series, the numerator comprising even numbers up to \(2n\) is divided by the factorial of \(2n\). Understanding the behavior of such terms as \(n\) approaches infinity is crucial to finding the convergence radius.
**Step-by-Step Analysis:**
1. **Identify the General Term:**
- The nth term is given by \(\frac{2 \cdot 4 \cdot 6 \cdot \ldots \cdot (2n)}{(2n)!} x^n\).
2. **Simplification:**
- Factorial simplification might be necessary to handle the series and calculate the limit as \(n\) tends to infinity.
3. **Apply a Convergence Test:**
- Use either the ratio test or root test to find the limit and, hence, determine the radius of convergence.
On completion, this process will yield the radius of convergence, denoted as \(R\). This indicates where the series converges in terms of the absolute value of \(x\).
**Note:** This explanation does not solve the problem but guides you on how to proceed for educational purposes.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6e511a64-2294-4edc-ba68-869688b4c27f%2F0c0f2e47-bd7d-4c28-adf3-5f55f52fb09c%2F1mq1kn9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Find the Radius of Convergence**
Consider the series:
\[
\sum_{n=1}^{\infty} \frac{2 \cdot 4 \cdot 6 \cdot \ldots \cdot (2n)}{(2n)!} x^n
\]
To determine the radius of convergence for this series, analyze the pattern and apply mathematical techniques such as the ratio test or root test. In this series, the numerator comprising even numbers up to \(2n\) is divided by the factorial of \(2n\). Understanding the behavior of such terms as \(n\) approaches infinity is crucial to finding the convergence radius.
**Step-by-Step Analysis:**
1. **Identify the General Term:**
- The nth term is given by \(\frac{2 \cdot 4 \cdot 6 \cdot \ldots \cdot (2n)}{(2n)!} x^n\).
2. **Simplification:**
- Factorial simplification might be necessary to handle the series and calculate the limit as \(n\) tends to infinity.
3. **Apply a Convergence Test:**
- Use either the ratio test or root test to find the limit and, hence, determine the radius of convergence.
On completion, this process will yield the radius of convergence, denoted as \(R\). This indicates where the series converges in terms of the absolute value of \(x\).
**Note:** This explanation does not solve the problem but guides you on how to proceed for educational purposes.
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