Can All Smooth Functions Near Zero Be Expressed by This Double Series?

[Ben Niehoff]

As you should know, a function can be smooth in some neighborhood and yet fail to be analytic. A canonical example is $\exp (-1/x^2)$ near $x = 0$. My question is this: suppose I want to express a given function as a double series,

$$f(x) = \sum_{m = 0}^\infty \sum_{n = 0}^\infty a_m x^m \exp \bigg( - \frac{b_n}{x^n} \bigg)$$
Is this series general enough to express all smooth functions in some open neighborhood of 0? And if not, what do the counterexamples look like?

This has some relationship to instantons in QFT.

 

[Ben Niehoff]

Never mind, I see that

$$f(x) = \exp ( - \exp (1/x^2) )$$
is a counterexample.

Carry on, then.