The Taylor series of a function f(x) around x a does not necessarily converge to the function itself anywhere except at x a.

Understanding Taylor Series Convergence

The Taylor series of the function relies on the function being analytic in the disk containing the point x a. To put it simply, f(x) is analytic if it can be represented as a power series around that point. If a function is not analytic, the Taylor series might not capture its behavior in a meaningful way, especially if the function has a special form like a flat function.

Example of a Flat Function

A notable example of a flat function is:

$$ f(x) begin{cases} e^{-1/x^2} amp; text{if } x eq 0 0 amp; text{if } x leq 0 end{cases} $$

As x approaches 0, (f(x)) approaches 0, and all its derivatives also tend to 0. Despite this, the Taylor series of f(x) at x 0 is identically zero, whereas the function itself is not. This illustrates that not all functions are adequately represented by their Taylor series expansions.

Cases of Convergence

For functions like (f(x) sin(x)), their Taylor series expansions converge to the function itself for all values of x. In such cases, the Taylor series of (f(x)) at x 0 (i.e., the Maclaurin series) is a perfect representation of the sine function within a large interval.

Complex Functions

For functions of a complex variable, (f(x) sin(x)) can also be represented as a power series, and it converges to the function within the disk of convergence. The radius of convergence is determined by the distance to the nearest singularity in the complex plane. In the case of the sine function, this convergence is within the entire complex plane, making it an entire function.

Convergence Conditions and Intervals

For real x, the convergence of the Taylor series can be analyzed by bounding the remainder term. The interval of convergence is where this remainder tends to zero as the degree of the polynomial increases to infinity. However, within this interval, the function can converge exactly to the original function.

For functions that can be represented as power series, the radius of convergence is typically explored to understand where this series converges to the function. If the function is an entire function (like the sine function), it can be expanded into a power series which converges everywhere.

In conclusion, while not every function's Taylor series converges to the function itself, for some specific functions, like the sine function, the series does converge and can be used to represent the entire function accurately.