Use the differentiation property on the Taylor series for to obtain a series for (1-¹). (Hint: You may have to differentiate more than once)
Use the differentiation property on the Taylor series for to obtain a series for (1-¹). (Hint: You may have to differentiate more than once)
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
![**Problem Statement:**
Use the differentiation property on the Taylor series for \(\frac{1}{1-x}\) to obtain a series for \(\frac{1}{(1-x)^3}\).
*(Hint: You may have to differentiate more than once)*
---
In this problem, you are tasked with using the properties of differentiation on the Taylor series expansion of the function \(\frac{1}{1-x}\) to find the expansion for the function \(\frac{1}{(1-x)^3}\).
**Steps to Consider:**
1. **Identify the Taylor Series of \(\frac{1}{1-x}\):**
The Taylor series of \(\frac{1}{1-x}\) is a geometric series:
\[
\frac{1}{1-x} = 1 + x + x^2 + x^3 + \ldots
\]
2. **Differentiate the Series:**
To find the series for \(\frac{1}{(1-x)^n}\) where \(n > 1\), you’ll typically differentiate the series multiple times. In this case, you will differentiate it twice.
3. **Apply the Differentiation Property:**
- Differentiate the series once to get the series for \(\frac{1}{(1-x)^2}\).
- Differentiate the resulting series again to find the series for \(\frac{1}{(1-x)^3}\).
These steps will guide you through using calculus-based properties to transform a known series into a more complex one through differentiation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9d258625-edd0-4f06-8dfd-6b1a087f7728%2Fddc5ce03-60c1-49b1-acd9-c7cf8e518786%2Fz8bze69_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Use the differentiation property on the Taylor series for \(\frac{1}{1-x}\) to obtain a series for \(\frac{1}{(1-x)^3}\).
*(Hint: You may have to differentiate more than once)*
---
In this problem, you are tasked with using the properties of differentiation on the Taylor series expansion of the function \(\frac{1}{1-x}\) to find the expansion for the function \(\frac{1}{(1-x)^3}\).
**Steps to Consider:**
1. **Identify the Taylor Series of \(\frac{1}{1-x}\):**
The Taylor series of \(\frac{1}{1-x}\) is a geometric series:
\[
\frac{1}{1-x} = 1 + x + x^2 + x^3 + \ldots
\]
2. **Differentiate the Series:**
To find the series for \(\frac{1}{(1-x)^n}\) where \(n > 1\), you’ll typically differentiate the series multiple times. In this case, you will differentiate it twice.
3. **Apply the Differentiation Property:**
- Differentiate the series once to get the series for \(\frac{1}{(1-x)^2}\).
- Differentiate the resulting series again to find the series for \(\frac{1}{(1-x)^3}\).
These steps will guide you through using calculus-based properties to transform a known series into a more complex one through differentiation.
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