**Taylor's Formula and displacement operator:** I (too often) see in papers (mathematical physics but a recent paper (a) by mathematicians also) the statement 

>**False belief 1 :** a) Let $D=\frac{d}{dx}$ be the derivation operator. Then, for all $f\in C^\infty(\mathbb{R})$, 
$$
e^{tD}[f](x)=f(x+t)
$$

which is false (take any $\phi\in C^\infty(\mathbb{R})$ with compact support, for instance).

> **False belief 2 :** b) In the same vein, for formal power series (“_if our object is formal then we do not have to ensure convergences_”). Let $S(x)\in \mathbb{R}[[x]]$ ($x$ is a formal variable) then for $t\in \mathbb{R}$, one has 
$$
e^{tD}[S](x)=S(x+t)
$$

which is false as we must have $t$ in the domain of convergence of $S$.
 
**Remarks** (i) The function $f\in C^\infty(\mathbb{R})$ is analytic over $\mathbb{R}$ iff 
$$
(\forall x\in \mathbb{R})(\exists R>0)(\forall t\in ]-R,R[)
(\sum_{n\geq 0}\frac{t^n}{n!}D^n[f](x)=f(x+t))\tag{1}\label{275319_1}
$$
(ii) Even if $f\in C^\omega(\mathbb{R})$, it can happen that the left hand side of eq. \eqref{275319_1} do not converge otherwise $f$ would be the restriction of an entire function (which e.g. $\frac{1}{1+x^2}$ is not, for example).  

(iii) Even if the LHS of \eqref{275319_1} converges for all $x,t\in \mathbb{R}$, $f$ need not be analytic. Consider the following function (classic in theory of distributions)  
$$
f(x)=0\text{ if } x\notin \mathopen]-1,1\mathclose[\text{ and } f(x)=e^{\frac{1}{1-x^2}} \text{ if } x\in \mathopen]-1,1[\mathclose.
$$
(iv) In the (b) case $S=\sum_{n\geq 0}n!\, x^n$ for example cannot be displaced.

(v) As @vanxoo [says](https://mathoverflow.net/posts/comments/1220689) false belief 2 could be cured by considering, in appropriate spaces, $t$ as a formal variable. This is the point of view of Bourbaki in _Functions of a real variable_ (see Ch VI _Generalized Taylor expansions_).