Problem with a taylor serie expansion

Thread starter Amaelle
Start date Jan 22, 2022
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Expansion Taylor

In summary, a Taylor series expansion is a representation of a function as an infinite sum of terms, used to approximate a function or find its value at a difficult point. It is commonly used in mathematics, physics, and engineering. However, there are some problems with using a Taylor series, such as its reliance on the number of terms and its convergence for certain functions. To improve accuracy, one can use more terms or higher-order approximations, or use numerical methods to determine and adjust for error. There are also alternative methods for approximating a function, such as polynomial interpolation, Fourier series, and numerical integration. The choice of method depends on the specific function and desired level of accuracy.

Jan 22, 2022

Amaelle

Homework Statement: Look at the image

Relevant Equations: Tayolr development

Greetings
https://www.physicsforums.com/attachments/295843

I really don´t agree with the solution
https://www.physicsforums.com/attachments/295846

as I calculated f_xy I got
f_xy=xye^xy
f(0,1)=0 so x(y-1) should not appear in the solution
am I wrong?

thank you!

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Jan 22, 2022

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Jan 22, 2022

Amaelle

thank you I solved the problem it was my wrong calculation!

What is a Taylor series expansion?

A Taylor series expansion is a mathematical representation of a function as an infinite sum of terms. It is used to approximate a function with a polynomial expression, making it easier to analyze and manipulate.

When is a Taylor series expansion useful?

A Taylor series expansion is useful when dealing with functions that are difficult to work with in their original form. By approximating the function with a polynomial expression, it becomes easier to calculate derivatives and integrals, and to make predictions about the behavior of the function.

What are some common problems with Taylor series expansions?

One common problem with Taylor series expansions is that they are only accurate within a certain interval, known as the convergence interval. Outside of this interval, the approximation may not be accurate. Another problem is that the accuracy of the approximation depends on the smoothness of the original function.

How can I determine the convergence interval of a Taylor series expansion?

The convergence interval of a Taylor series expansion can be determined by using the ratio test or the root test. These tests involve taking the limit of the ratio or root of successive terms in the series. If the limit is less than 1, the series converges within that interval.

What are some applications of Taylor series expansions in science?

Taylor series expansions are used in many areas of science, including physics, engineering, and economics. They are used to approximate solutions to differential equations, to model complex systems, and to analyze the behavior of physical phenomena. They are also used in computer algorithms for data analysis and prediction.