Difficult# Convergence of Taylor Series to the Original Function

SVCALC-4W9I1U

Consider the function:

$$f(x)=\begin{cases} e^{-1/x}\quad x>0 \\\ 0\quad\quad x\leq0\end{cases}$$

It can be shown that $f(x)$ is infinitely differentiable. Let $a$ be a positive number.

Does the Taylor series of $f(x)$ at $x=0$ converge for all $x$ values in the interval $(-a,a)$, and does $f(x)$ equal its Taylor series on $(-a,a)$?

*(The answer to this question doesn't depend on the exact value of $a$. To answer this question, start by computing the derivatives of $f(x)$ at $0$.)*