Is a germ equivalent to an infinite jet?

Not all smooth functions are analytic, as it is well known, so they in general cannot be represented as a power series.

If we restrict our attention to analytic functions, then a specification of the values of all derivatives of a function at a point will give us the function.

My question is essentially about how much information is contained in knowing all derivatives of a smooth but not necessarily analytic function at a point.

In particular, let $(E, \pi, M) $ be a fibred manifold and let $\Gamma_x(E) $ denote the space of germs of smooth sections at $x\in M$. A germ contains information about the section in an arbitrarily small open neighborhood of $x$. However, practically thinking, the only definite values I can associate to a germ is the value of a representative section at $x$, and the values of its derivatives at the same point to arbitrary orders.

On the other hand, points of the infinite jet space $J_x^\infty(E) $ literally consist of values of a section and derivatives up to all orders at $x$. Nonetheless I feel that the germ space might contain more "nonlocal" information than the infinite jet space.

So the question is, how are $J_x^\infty(E) $ and $\Gamma_x(E) $ related?

Most of the relevant behavior already occurs in the simple case where $M = \Bbb R$ and $E \to \Bbb R$ is the trivial bundle, so $\Gamma^\infty_x(E)$ is just the space ${\mathcal C}^\infty_x$ of germs of smooth functions $\Bbb R \to \Bbb R$ at $x$, and $J^\infty_x(E)$ is just the usual jet space $J^\infty_x(\Bbb R, \Bbb R)$.

For any representative $f$ of a germ $[f]$ at $x_0$ we can compute all of the derivatives $f^{(k)}(x)$ of $f$ at $x$, that is, the $\infty$-jet $j^{\infty}_x(f) \in J^{\infty}_x$ of $f$ at $x$, and since the definition of a derivative is local this jet is independent of the choice of representative $f$. So, we have a well-defined map $$\Pi : \Gamma_x(E) \to J^{\infty}_x(E) .$$ It follows from the definitions of germs and jets that this map is surjective. Moreover, Borel's Lemma tells us that every formal power series at $x$ is the Taylor series of some smooth function at $x$, so we can identify, I think, $J^\infty_x(E)$ with the space $\Bbb R[[x]]$ of formal power series.

On the other hand, as Hagen von Eitzen recalled in the comments, there are nonzero smooth functions with zero $\infty$-jet, like the classic example $x \mapsto \exp\left(-x^{-2}\right)$ (with the singularity removed), so $\Pi$ is not injective. Thus, a germ at a point does contain strictly more information than the corresponding $\infty$-jet at that point, but to be sure this information is local, since the definition of a germ is.

Conversely, the power series of a real-analytic function $f$ at some point converges to that function in some neighborhood of that point, so we can recover the germ $[f]$ from its $\infty$-jet. So, if we denote by $\mathcal{C}^\omega_x$ the set of germs of real-analytic functions at $x$, the restriction $$\Pi\vert_{\mathcal{C}^\omega_x} : \mathcal{C}^\omega_x \to J^\infty_x(E)$$ of $\Pi$ to that set is injective, and we can thus identify $\mathcal{C}^\omega_x$ with a (proper) subspace of $J^\infty_x(E)$. On the other hand, since there are formal power series that do not converge on any open set (apply the Ratio Test to the series $\sum_{n = 0}^\infty n! x^n$) this restriction is not surjective.

To talk about real-analyticity for general fiber bundles $E \to M$, $E$ and $M$ must come equipped with (compatible) real-analytic atlases. Since our considerations are local, we may as well work in an (analytic) fiber chart, reducing the problem to understanding the behavior of jets of functions $\Bbb R^m \to \Bbb R^n$, so not much changes from the above toy case. (I haven't thought about jets of fibered manifolds, but I don't see immediately that anything that goes wrong in that case.)

Remark It's natural to ask how we can describe the information contained in a germ but not its corresponding jet. Again in the toy case, the map $\Pi$ fits into a short exact sequence, $$0 \to \ker \Pi \to \mathcal{C}^\infty_x \stackrel{\Pi}{\to} J^\infty_x \to 0 .$$ By definition, $\ker \Pi$ is just the space of germs whose corresponding jet is the zero jet. A choice of complement $N$ of $\ker \Pi$ in $C^\infty_x$ (that is, a choice of splitting of the s.e.s.) induces a decomposition $\mathcal{C}^\infty_x = \ker \Pi \oplus N \cong \ker \Pi \oplus J^\infty_x$, and then the information contained in a germ $[f]$ but not the corresponding jet $j^\infty_x f$ is encoded (in a way that depends on the choice of $N$) as the projection of $[f]$ to $\ker \Pi$. I don't think, however, that there is a natural choice of complement.

On the other hand, there is a natural subspace of $\mathcal{C}^\infty_x$ in which there is a natural complement of the kernel of the restriction of $\Pi$. Define $$\mathcal D_x := (\Pi\vert_{\mathcal{C}^\infty_x})^{-1} (\Pi(\mathcal{C}^\omega_x)) .$$ Unwinding definitions, we see that $\mathcal D_x$ is precisely the space of germs whose power series at $x$ converges on some open subset containing $x$. This gives rise to a longer exact sequence, $$0 \to \ker \Pi\vert_{\mathcal D_x} \to \mathcal D_x \stackrel{\Pi_{\mathcal D_x}}{\to} J^\infty_x \to J^\infty_x / \mathcal D_x \to 0 .$$ Then, $C^\omega_x$ is natural complement to $\ker \Pi\vert_{\mathcal D_x}$: The splitting map $\mathcal D_x \to \ker \Pi\vert_{\mathcal D_x}$---which again per the above we interpret as the map sending a germ to the information contained therein but not in the corresponding jet---is just $$[f] \mapsto [f] - (\Pi\vert_{\mathcal{C}^\omega_x})^{-1} (\Pi([f])) .$$ Informally, this map subtracts from a germ the germ of the function to which the power series of the germ converges, leaving a germ with zero power series.