Find the power series for f(x) = ln(1 – x) centered at æ = term integration or differentiation. O by using term-by- |
Find the power series for f(x) = ln(1 – x) centered at æ = term integration or differentiation. O by using term-by- |
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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![**Problem Statement:**
Find the power series for \( f(x) = \ln(1-x) \) centered at \( x = 0 \) by using term-by-term integration or differentiation.
\[
f(x) = \sum_{n=1}^{\infty}
\]
**Explanation:**
You are tasked with finding the power series representation of the natural logarithm function \( f(x) = \ln(1-x) \) about the center \( x = 0 \). This involves expressing \( f(x) \) as a series of powers of \( x \). To achieve this, you can utilize either term-by-term integration or differentiation, which are techniques often used to derive new series from known ones.
The power series format is given as:
\[
f(x) = \sum_{n=1}^{\infty} a_n x^n
\]
where \( a_n \) represents the coefficients of the series to be determined.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F62b092f3-63db-4d93-982f-67d0473d8e68%2F0180114b-0778-4814-9a31-3fef9bfd5f9b%2Fuigbini_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the power series for \( f(x) = \ln(1-x) \) centered at \( x = 0 \) by using term-by-term integration or differentiation.
\[
f(x) = \sum_{n=1}^{\infty}
\]
**Explanation:**
You are tasked with finding the power series representation of the natural logarithm function \( f(x) = \ln(1-x) \) about the center \( x = 0 \). This involves expressing \( f(x) \) as a series of powers of \( x \). To achieve this, you can utilize either term-by-term integration or differentiation, which are techniques often used to derive new series from known ones.
The power series format is given as:
\[
f(x) = \sum_{n=1}^{\infty} a_n x^n
\]
where \( a_n \) represents the coefficients of the series to be determined.
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