Show that if f has a power series representation (expansion) at a, that is, if f(x)=EG, (x-a)" = c, +G (x-a)+c,(x-a) +c, (x-a)'+.. n=0 or x-a< R, then its coefficients are given by the formula orit zordioriw orrianotob ot teoT lon orit eal %3D n! "his series is called a Taylor series (ifa=0 it is called a Maclaurin series).
Show that if f has a power series representation (expansion) at a, that is, if f(x)=EG, (x-a)" = c, +G (x-a)+c,(x-a) +c, (x-a)'+.. n=0 or x-a< R, then its coefficients are given by the formula orit zordioriw orrianotob ot teoT lon orit eal %3D n! "his series is called a Taylor series (ifa=0 it is called a Maclaurin series).
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Taylor Series Expansion
#### Power Series Representation
If a function \( f \) has a power series representation (expansion) at \( a \), it is expressed as:
\[
f(x) = \sum_{n=0}^{\infty} c_n (x - a)^n = c_0 + c_1 (x - a) + c_2 (x - a)^2 + c_3 (x - a)^3 + \cdots
\]
For \(|x - a| < R\), the coefficients \( c_n \) are given by the formula:
\[ c_n = \frac{f^{(n)}(a)}{n!} \]
This series is called a **Taylor series**. If \( a = 0 \), it is called a **Maclaurin series**.
The Taylor series provides a powerful tool for approximating functions using polynomials, especially around a point \( a \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffa41b47d-a27d-4623-80de-b3b064f82b52%2F7240338e-829a-43c5-a5f6-e3dd878524e5%2Fqsazi0k_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Taylor Series Expansion
#### Power Series Representation
If a function \( f \) has a power series representation (expansion) at \( a \), it is expressed as:
\[
f(x) = \sum_{n=0}^{\infty} c_n (x - a)^n = c_0 + c_1 (x - a) + c_2 (x - a)^2 + c_3 (x - a)^3 + \cdots
\]
For \(|x - a| < R\), the coefficients \( c_n \) are given by the formula:
\[ c_n = \frac{f^{(n)}(a)}{n!} \]
This series is called a **Taylor series**. If \( a = 0 \), it is called a **Maclaurin series**.
The Taylor series provides a powerful tool for approximating functions using polynomials, especially around a point \( a \).
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