This is a soft question.
As I understand it, Borel's lemma implies that, given any sequence of real numbers $(a_n)_{n\geq 0}$, the formal power series
$$ \sum_{n\geq 0} a_nx^n $$
is the Taylor expansion at the origin of some smooth function $f:\mathbb{R}\rightarrow\mathbb{R}$.
Apply this with a sufficiently fast-growing sequence $a_n$ such that the radius of convergence of the series is zero, for example $a_n = n!$, yielding some smooth function $f$ with $(d^nf/dx^n)_{x=0} = n!^2$.
I am having trouble understanding $f$. Evidently, it cannot be analytic. Fine, there are many smooth functions that are not analytic. But the main examples of such in my mind (e.g. $e^{-1/x^2}$ at $x=0$) are locally approximated by analytic functions, in the sense that at any given point there is an analytic function with the same power series expansion. The $f$ in question is smooth but is not locally approximated by an analytic function (at $x=0$, the relevant point) in this sense.
What does $f$'s too-fast-growing Taylor expansion force it to look like around $x=0$?
Apologies again that this is a soft question. Something is bothering me, and I'm not exactly sure what type of information would resolve the dissonance. (That said, I could imagine a complete/precise answer consisting of pointing out to me something important I am missing.)
Remark: Another framework for what's going on. Borel's lemma asserts that the map $C^\infty(\mathbb{R})\rightarrow \mathbb{R}[[x]]$ sending a smooth function to its Taylor expansion at $0$ is surjective. In fact, this map is well-defined on the ring $C_{x=0}^\infty(\mathbb{R})$ of germs of smooth functions at $x=0$, and $C_{x=0}^\infty(\mathbb{R})\rightarrow \mathbb{R}[[x]]$ is surjective by Borel's lemma. Now we can restrict this map to the germs of analytic functions, each defined on some (arbitrarily small) neighborhood of $x=0$. Then, the above reasoning is saying it stops being surjective (although the image is evidently still dense since germs of even just polynomials map to polynomials in $\mathbb{R}[[x]]$, and these are dense). What do the functions representing smooth germs that are not analytic germs look like? (Note that the answer here is not what I'm looking for. In that question, the key point was that an analytic germ may not extend to a global analytic function [and therefore you can't construct the whole ring of analytic germs by beginning with the global analytic functions]. Here I am interested in the smooth germs that do not come from analytic germs at all.)
