Chapter 9.9, Problem 14ES

The Purpose of this exercise is to show that the Taylor series of a function $f$ may possibly converge to a value different from $f(x)$ for certain values of $x.$ Let $f(x)=\left\{\begin{array}{l}{e}^{-1/x^2,}\\ 0,\end{array}\right.\begin{array}{c}x\ne 0\\ x=0\end{array}$

(a) Use the definition of a derivative to show that $f\text{'}(0)=0.$

(b) With some difficulty it can be shown that if $n\ge 2,\text{then}{f}^{(n)}(0)=0.$ Accepting this fact, show that the Maclaurin series of $f$ converges for all $x$ but converges to $f(x)$ only at $x=0.$