Given The for Taylw Seni 1-x +x²-x³+ .eer %3D %3D Oltaii The Seris for Seven lalixl 7メリツ
Given The for Taylw Seni 1-x +x²-x³+ .eer %3D %3D Oltaii The Seris for Seven lalixl 7メリツ
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
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![**Understanding the Taylor Series and Obtaining the Series for \( \ln(|1+x|) \)**
**Taylor Series Expansion Explanation:**
Given the Taylor Series expansion for the function \( \frac{1}{1+x} \):
\[ \frac{1}{1+x} = 1 - x + x^2 - x^3 + \cdots \]
This can be represented as an infinite sum notation:
\[ \frac{1}{1+x} = \sum_{n=0}^{\infty} (-1)^n x^n \]
where \( n \) ranges from 0 to infinity. Each term in the series alternates in sign and increases in the power of \( x \).
**Task:**
Now, obtain the series for \( \ln(|1+x|) \).
---
In the context of an educational website, the above transcription will help students understand the given Taylor Series expansion for \( \frac{1}{1+x} \) and guide them towards finding the series expansion for \( \ln(|1+x|) \) through manipulation of the given series.
**Diagram Explanation:**
There are no diagrams or graphs in this image. The focus is on the mathematical equations and the task to derive the series for \( \ln(|1+x|) \).
---
The above content is informative and precise for educational purposes, aiding in the conceptual understanding of series expansions in calculus.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faaa612e4-bb6a-4a7c-9bcb-a4980796bdd1%2F1ebcf11d-5c7e-4037-8cd4-ebfc2e92386d%2Fu0c81j.jpeg&w=3840&q=75)
Transcribed Image Text:**Understanding the Taylor Series and Obtaining the Series for \( \ln(|1+x|) \)**
**Taylor Series Expansion Explanation:**
Given the Taylor Series expansion for the function \( \frac{1}{1+x} \):
\[ \frac{1}{1+x} = 1 - x + x^2 - x^3 + \cdots \]
This can be represented as an infinite sum notation:
\[ \frac{1}{1+x} = \sum_{n=0}^{\infty} (-1)^n x^n \]
where \( n \) ranges from 0 to infinity. Each term in the series alternates in sign and increases in the power of \( x \).
**Task:**
Now, obtain the series for \( \ln(|1+x|) \).
---
In the context of an educational website, the above transcription will help students understand the given Taylor Series expansion for \( \frac{1}{1+x} \) and guide them towards finding the series expansion for \( \ln(|1+x|) \) through manipulation of the given series.
**Diagram Explanation:**
There are no diagrams or graphs in this image. The focus is on the mathematical equations and the task to derive the series for \( \ln(|1+x|) \).
---
The above content is informative and precise for educational purposes, aiding in the conceptual understanding of series expansions in calculus.
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