Use that result to find the power series representation of the integral ſ 1+x¹ dx and determine its radius of convergence. sion to determine its
Use that result to find the power series representation of the integral ſ 1+x¹ dx and determine its radius of convergence. sion to determine its
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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![**Educational Explanation: Writing Functions as Power Series**
**Example 4: Understanding the Importance of Power Series Expressions**
i) Here is an interesting fact: The antiderivative of the function \( f(x) = \frac{1}{1+x^4} \) does not exist in a standard elementary form. For now, trust this statement, though you're welcome to investigate further on your own.
ii) Sometimes in calculus, we need to evaluate integrals like: \(\int \frac{1}{1+x^4} \, dx\).
From class, you might remember:
\[
\frac{1}{1+x} = \sum_{n=0}^{\infty} (-1)^n x^n = 1 - x + x^2 - x^3 + \ldots
\]
This series converges when \(|x| < 1\).
**Task:**
Use this series to find the power series representation of the integral \(\int \frac{1}{1+x^4} \, dx\), and determine its radius of convergence.
**Conclusion:**
Even though we cannot easily find the antiderivative of the function \( f(x) \), expressing it as a power series allows us to evaluate its integral effectively. This is a powerful technique to solve complex problems in calculus!](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff675898d-3fe8-40cd-9afe-fa982eda0648%2F000e221e-9672-44f0-82bd-cc784ca5bf1c%2F87jrbv_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Educational Explanation: Writing Functions as Power Series**
**Example 4: Understanding the Importance of Power Series Expressions**
i) Here is an interesting fact: The antiderivative of the function \( f(x) = \frac{1}{1+x^4} \) does not exist in a standard elementary form. For now, trust this statement, though you're welcome to investigate further on your own.
ii) Sometimes in calculus, we need to evaluate integrals like: \(\int \frac{1}{1+x^4} \, dx\).
From class, you might remember:
\[
\frac{1}{1+x} = \sum_{n=0}^{\infty} (-1)^n x^n = 1 - x + x^2 - x^3 + \ldots
\]
This series converges when \(|x| < 1\).
**Task:**
Use this series to find the power series representation of the integral \(\int \frac{1}{1+x^4} \, dx\), and determine its radius of convergence.
**Conclusion:**
Even though we cannot easily find the antiderivative of the function \( f(x) \), expressing it as a power series allows us to evaluate its integral effectively. This is a powerful technique to solve complex problems in calculus!
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