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Q. 9

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Calculus
Found in: Page 679
Calculus

Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861

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Short Answer

What is a difference between a Taylor polynomial and the Taylor series for a function f at a point x0?

So, a Taylor polynomial of degree n is approximation to the function f at x= x0.It is possible for a Taylor series for a function f to converge to f on the interval ofconvergence for the series. In other words, a Taylor polynomial is an alternate wayto represent the function of the interval of convergence. But this approximation byTaylor polynomial of the function f improves as the degree of the polynomial increases.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1. Given information is:

A function f with Taylor polynomial and the Taylor series

Step 2. Taylor polynomial 

For any function f with derivative of order n, then the Taylorpolynomial of degree n at x=x0 is, Pn(x=x0)=f(x0) +f'(x0)(x-x0)+f''(x0)2!(x-x0)2+....+fn(x0)n!(x-x0)nThe general form Taylor polynomial of degree n of the function f is:Pn(x)=∑k=0nfk(x0)k!(x-x0)k

Step 3. Taylor Series

For any function f with derivative of order n, then the Taylorseries at x=x0 is, Pn(x=x0)=f(x0) +f'(x0)(x-x0)+f''(x0)2!(x-x0)2+.....The general form Taylor series of the function f is:Pn(x)=∑k=0∞fk(x0)k!(x-x0)k

Step 4. Result

So, a Taylor polynomial of degree n is approximation to the function f at x= x0.It is possible for a Taylor series for a function f to converge to f on the interval ofconvergence for the series. In other words, a Taylor polynomial is an alternate wayto represent the function of the interval of convergence. But this approximation byTaylor polynomial of the function f improves as the degree of the polynomial increases.

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