How to prove Taylor's expansion for multivariable function ?

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Taylor's expansion for multivariable functions involves expressing a function as a series of terms based on its derivatives at a specific point. To prove this expansion, the function must be infinitely differentiable in the neighborhood of that point. The discussion highlights that while a function can have a Taylor series that converges, it may not equal the function itself everywhere, as illustrated by the example of f(x) = e^{-1/x^2}. The focus is on determining the form of the Taylor series for well-behaved multivariable functions and the conditions necessary for its validity. Understanding these concepts is crucial for applying Taylor's expansion effectively in multivariable calculus.
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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 f(x)= e^{-1/x^2} is x\ne 0, 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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