How to prove Taylor's expansion for multivariable function ?

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SUMMARY

The discussion centers on proving Taylor's expansion for multivariable functions. It clarifies that Taylor's series requires the function to be infinitely differentiable for its series to exist. A specific example provided is the function f(x) = e^{-1/x^2} for x ≠ 0, which has a Taylor series that converges uniformly but does not equal the function except at x = 0. The main inquiry is about the form of Taylor's series for well-behaved multivariable functions and the proof of its validity.

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  • Understanding of Taylor's series and its definition.
  • Knowledge of multivariable calculus concepts.
  • Familiarity with the concept of uniform convergence.
  • Experience with infinitely differentiable functions.
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  • Study the definition and properties of Taylor's series in multivariable calculus.
  • Learn about uniform convergence and its implications for Taylor series.
  • Explore examples of infinitely differentiable functions and their Taylor series.
  • Investigate the conditions under which a Taylor series converges to the original function.
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Mathematicians, students of multivariable calculus, and anyone interested in the theoretical foundations of Taylor's expansion in higher dimensions.

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As mentioned in the title~

Could anyone give me a hint or an idea ?

Thanks~
 
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What do you mean by "prove Taylor's expansion"? Taylor's expansion (by which I take it you mean writing a Taylor's series for the function) is defined to be a specific thing. Do you mean "proving the function is equal to its Taylor's series on the interval of convergence"? If so, what restrictions are you putting on the function. Of course, you must have f infinitely differentiable in order that its Taylor's series exist but there exist infinitely differentiable functionsm, whose Taylor's series converge for all x but that are not equal to their Taylor series anywhere except at the base point. One such is [itex]f(x)= e^{-1/x^2}[/itex] is [itex]x\ne 0[/itex], 0 if x= 0. Its Taylor's series, about x= 0, is identically 0 so converges uniformly for all x but is not equal to f(x) anywhere except at x= 0.
 
Thank your warning~

Let me restate my question:

What's the form of Taylor's series about a "well-behaved" multivariable function ?

And~how to prove it ?
 

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