Power series expansion of $e^{-1/x^2}$ at a point different from 0

Question

The function $f(x)=e^{-1/x^2}$ ($f(0)=0$) does not have a power series expansion at $z_0=0$.

Now my question: Is there a power series for $f$ centered at $z_0\neq0$ with convergence radius greater than $|z_0|$?

Answer 1

If there was, then you could use it to construct a power series expansion of $f$ centered at $0$ (by using, e.g., the analyticity of holomorphic functions). So there isn't.