Confusion with the Taylor's Inequality

Thread starter yupenn
Start date Aug 15, 2009
Tags

Confusion Inequality

In summary, Taylor's Inequality is a mathematical tool used in calculus to estimate the error between a function and its Taylor polynomial approximation. It is used to determine the accuracy of the approximation by providing an upper bound on the difference between the actual value of the function and its approximation. The formula for Taylor's Inequality is |f(x) - P_n(x)| ≤ M * (x-a)^(n+1) / (n+1)!, where f(x) is the function, P_n(x) is the Taylor polynomial approximation of degree n, a is the center of the approximation, and M is the maximum value of the (n+1)th derivative of the function on the interval [a,x]. Its

Aug 15, 2009

yupenn

Taylor's Inequality states:
if |f n+1(x)|<=M
then
|Rn(x)|<=M*|x-a|^(n+1)/(n+1)!
however,
Rn(x)=f n+1(x)*|x-a|^(n+1)/(n+1)!+...
it seems |Rn(x)|>=M*|x-a|^(n+1)/(n+1)! when |f n+1(x)|=M

the inequality is wrong??

Physics news on Phys.org

Aug 18, 2009

HallsofIvy

Science Advisor

Homework Helper

You seem to be assuming that all derivatives of f are positive which is NOT generally true.

1. What is Taylor's Inequality?

Taylor's Inequality is a mathematical tool used in calculus to estimate the error between a function and its Taylor polynomial approximation. It provides a bound on the difference between the actual value of the function and its approximation.

2. How is Taylor's Inequality used?

Taylor's Inequality is used in calculus to approximate functions using a polynomial. It helps to determine the accuracy of the approximation by providing an upper bound on the error between the actual function and its approximation.

3. What is the formula for Taylor's Inequality?

The formula for Taylor's Inequality is |f(x) - P_n(x)| ≤ M * (x-a)^(n+1) / (n+1)!, where f(x) is the function, P_n(x) is the Taylor polynomial approximation of degree n, a is the center of the approximation, and M is the maximum value of the (n+1)th derivative of the function on the interval [a,x].

4. What is the significance of Taylor's Inequality?

Taylor's Inequality allows us to approximate functions with polynomials and determine the accuracy of the approximation. It is an essential tool in calculus and is used in many applications, such as physics, engineering, and economics.

5. Can Taylor's Inequality be applied to all functions?

Taylor's Inequality can be applied to all functions that are differentiable on the interval of approximation. However, the accuracy of the approximation may vary depending on the behavior of the function and the degree of the Taylor polynomial used.