power series expansion of the function $ e^{-\frac {1}{x^2}} $ about $0 $ I know that the function $e^{-\frac {1}{x^2}}$ is infinite  differentiable in a neighborhood of $0$ in $\mathbb{R}$. Does a power series representation exist for this function? If it does not exist, please provide some explanation. 

Let $f(x)=e^{-1/x^2}$ if $x\ne 0$, and let $f(0)=0$. All the derivatives of $f$ at $0$ are $0$.
The function $f$  has a delightfully simple Maclaurin series. The series happens to converge to $f$ only at $x=0$.
