The conditions under which the Taylor polynomial $P_n(x)$ will converge to $f(x)$

Question

Background

It is well-known that for the following function:

$$f(x) = \begin{cases} e^{\frac{-1}{x^2}} & \text{if} & x \neq 0 \\ 0 & \text{if} & x = 0 \end{cases} $$

the Taylor polynomial does not converge to $f(x)$ as $n\to \infty$, although $f^{(k)}(0)$ exists and it is 0 for all $k\geq 0$. The problem is with the general conditions under which a Taylor polynomial will converge to the function.

Problem

In general, what conditions exist (or do they exist at all) such that the Taylor polynomial of a function will converge to the function as $n\to \infty$?

Answer 1

One such condition is given by Bernstein's theorem: if $(\forall x\in D_f)(\forall n\in\mathbb{N}):f^{(n)}(x)\geqslant0$, then the Taylor series of $f$ converges pointwise to $f(x)$ in the neighborhood of every point of $D_f$.