Proof for a Taylor series of a function exactly representing that function as you expand the series to infinite terms?

Question

I understand that a taylor series is used to approximate a given function at a point to any required degree of accuracy, I have not however understood why when we take the taylor series of a function at point, expanding to infinite terms it perfectly represents the function over the entire domain of the function. Is there a mathematical proof for why this results from approximating a function and its infinite derivatives at a single point

Answer 1

Your intuition is correct.

A Taylor series is a formal representation of a function to all orders of approximations including the convergent infinite series; but generally, the series converges in a finite circle of convergence only.

Its clear, that normally, for large x, the terms $a_n x^n$ may diverge, except for series where $|a_n|\to 0$ faster, than $|x^n|\to \infty$ for any $x$.

This class of functions is called "entier" and includes polynomials, of course, and all exponential and trigonometric functions. By the polynomial algebra we take the fact for granted, that polynomials have completely different series representations dependent on the point of expansion in powers of $(x-a)^n$

In the same way the domain of definition of a series with a finite circle of convergency in the complex plane is not restricted to its primary circle.

With the exception of singular points, at any point, another Taylor series with another circle of convergency can established by the sole condition, that $z\to f(z)$ is complex differentiable in an open neighborhood of the point. These circles can overlap and in this way cover all of the complex plane with the exception of some singular points and eventually arbitrary cuts between them, if the values defined on chains o f circles don't coincide on different paths.

"Pathological" in nearly all cases are inverse function like the roots of polynomials and the log-like function as inverse of the exponentials, because functions can be inverted only in a domain were it is a 1-1 map. .