Why does perturbation theory involve a Taylor series rather than a Laurent series? Perturbation theory in the QM and the QFT is usually explained in terms of a small parameter expansion $\epsilon$ and expanded in a Taylor series.
$$O(t)=O_0(t)+\epsilon O_1(t)+\epsilon^2 O_t(t)+...$$
However, if we recall the course in transition from the real calculus to complex analysis, there is an emphasized transition from a Taylor series to Laurent series, where the terms of the divergent order were included, i.e.
$$\epsilon^{-1} O_{-1}(t)+\epsilon^{-2} O_{-2}(t)+...$$
But though complex numbers are popularly used, the perturbation approach is built from the Taylor series (at least in the QM and the QFT), and rarely were Laurent series used (except perhaps special branches like strings). This seems to be a bit counter intuitive from a mathematical perspective, and seemingly indicates some structures or assumptions built in the derivation of the theories.
Why does perturbation theoy involve a Taylor series rather than a Laurent series in QM and QFT?
 A: This is a nice question. However, notice that the idea behind perturbation theory is being able to write $O(t)$ as $O_0(t) \ +$ higher order terms, that should be much less relevant, i.e.,
\begin{equation}
|\varepsilon^k O_k(t)|\ll |O_0(t)|, \forall k>0
\end{equation}
Thus you are interested mostly in the case $\varepsilon \to 0$. Then it doesn't make much sense to consider Laurent series: you are just introducing unnecessary divergences.
A: Generically, as has been said, it is nice if the observable you're trying to compute is well defined as $\epsilon\to0$.
However, there are certain phenomena which can happen which go in the direction you mention. Firstly, when $\epsilon$ is the dimensional regularisation parameter (a regularisation scheme when we deform away from integer 4 dimensions to $4+\epsilon$), infrared divergences of theories which suffer from them like Yang-Mills do show up as higher order terms in $1/\epsilon$, but this is the signal that something goes bad with the model.
Furthermore, a very physical phenomenon which generalises the Taylor expansion you mention, and occur in QM and QFT, is the presence of non-perturbative effects (instantons, etc.) which show up as $\exp(-1/g)$ effects. They are thus invisible in the Taylor expansion near weak coupling $g=0$ but do affect the summability of the perturbative series.
