Convergence of Taylor series in several variables

Thread starter zetafunction
Start date Aug 23, 2010
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Convergence Series Taylor Taylor series Variables

In summary, the Taylor series in several variables is a mathematical representation of a function using its derivatives at a specific point. It converges when the function is analytic, and the approximation becomes more accurate as more terms are added. Absolute convergence means the series always converges to the same value, while convergence refers to the accuracy of the approximation. The Cauchy-Hadamard theorem can be used to determine the radius of convergence. The Taylor series in several variables can only be used to approximate analytic functions.

Aug 23, 2010

zetafunction

where do a multiple Taylor series converge ??

i mean if given a function [tex] f(x,y) [/tex] can i expand this f into a double Taylor series that will converge on a rectangle ? for example , if one can ensure that it converges for |x| <1 and |y| <1

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Aug 23, 2010

mathman

Science Advisor

It depends very much on what f is.

1. What is the Taylor series in several variables?

The Taylor series in several variables is a mathematical representation of a function in terms of its derivatives at a specific point. It is a way to approximate a function using a polynomial series.

2. How does the Taylor series in several variables converge?

The Taylor series in several variables converges when the function is analytic, meaning it can be represented by a convergent power series in a neighborhood of the point of expansion. This means that as more terms are added to the series, the approximation of the function becomes more accurate.

3. What is the difference between convergence and absolute convergence in Taylor series in several variables?

Convergence refers to the idea that as more terms are added to the series, the approximation of the function becomes more accurate. Absolute convergence, on the other hand, means that the series always converges to the same value regardless of the order in which the terms are added.

4. How do you determine the radius of convergence for a Taylor series in several variables?

The radius of convergence for a Taylor series in several variables can be determined by using the Cauchy-Hadamard theorem. This theorem states that the radius of convergence is equal to the inverse of the supremum of the set of points where the series converges.

5. Can the Taylor series in several variables be used to approximate any function?

No, the Taylor series in several variables can only be used to approximate functions that are analytic. If a function is not analytic, then the Taylor series may not converge or may converge to a different function.