Differentiate the given series expansion of f term-by-term to obtain the corresponding series expansion for the derivative of f. If f(x) = = 1 1 + 2x f'(x) = Σ n=1 = Σ (-1)"2"x" n=0
Differentiate the given series expansion of f term-by-term to obtain the corresponding series expansion for the derivative of f. If f(x) = = 1 1 + 2x f'(x) = Σ n=1 = Σ (-1)"2"x" n=0
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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6.2.5
![**Transcription of Mathematical Content**
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**Title:** Differentiation of Series Expansion
**Description:**
**Problem Statement:**
Differentiate the given series expansion of \( f \) term-by-term to obtain the corresponding series expansion for the derivative of \( f \).
**Given Function:**
\[ f(x) = \frac{1}{1+2x} = \sum_{n=0}^{\infty} (-1)^n 2^n x^n \]
**Objective:**
Find the series expansion of the derivative \( f'(x) \).
**Expression for the Derivative:**
\[ f'(x) = \sum_{n=1}^{\infty} \]
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**Explanation:**
The task involves differentiating the given power series of the function \( f(x) \), which is expressed as an infinite sum. The goal is to derive the series expansion of \( f'(x) \) by differentiating each term in the series. The notation used in the problem describes a common method for representing infinite series, where \( n \) is the index of summation starting at zero and extending to infinity.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6e511a64-2294-4edc-ba68-869688b4c27f%2Fd832724c-f9f0-4d80-95da-fad4868f880d%2F618g5hn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Transcription of Mathematical Content**
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**Title:** Differentiation of Series Expansion
**Description:**
**Problem Statement:**
Differentiate the given series expansion of \( f \) term-by-term to obtain the corresponding series expansion for the derivative of \( f \).
**Given Function:**
\[ f(x) = \frac{1}{1+2x} = \sum_{n=0}^{\infty} (-1)^n 2^n x^n \]
**Objective:**
Find the series expansion of the derivative \( f'(x) \).
**Expression for the Derivative:**
\[ f'(x) = \sum_{n=1}^{\infty} \]
---
**Explanation:**
The task involves differentiating the given power series of the function \( f(x) \), which is expressed as an infinite sum. The goal is to derive the series expansion of \( f'(x) \) by differentiating each term in the series. The notation used in the problem describes a common method for representing infinite series, where \( n \) is the index of summation starting at zero and extending to infinity.
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