Convergence of Taylor series in several variables

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SUMMARY

The discussion centers on the convergence of multiple Taylor series, specifically for functions of two variables, such as f(x,y). It is established that a double Taylor series can converge within a specified region, such as a rectangle defined by |x| < 1 and |y| < 1. However, the convergence is highly dependent on the properties of the function f itself, indicating that not all functions will exhibit this behavior.

PREREQUISITES
  • Understanding of Taylor series expansion
  • Familiarity with multivariable calculus
  • Knowledge of convergence criteria for series
  • Basic concepts of function behavior in defined regions
NEXT STEPS
  • Research convergence criteria for multivariable Taylor series
  • Study specific functions that exhibit double Taylor series convergence
  • Explore the implications of function properties on series convergence
  • Learn about regions of convergence in multivariable calculus
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Mathematicians, students of calculus, and anyone interested in the behavior of multivariable functions and their series expansions.

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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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It depends very much on what f is.
 

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