By "transcendentally small terms" I mean corrections to a perturbation that are smaller than all polynomial orders. They are generally exponential in form. I would have linked to a Wikipedia page, but I can't seem to find the term used very often.
Usually perturbation theory is not concerned with small corrections beyond all orders, but if the leading correction is exponentially suppressed then it (in principle) becomes relevant.
Take, as an elementary example, an integral expression from solid state physics: $$ n = \int^{\infty}_{0}\frac{g(\epsilon)}{e^{(\epsilon-\mu)\beta}+1}\text{d} \epsilon $$
which computes electron density.
In the limit as $\beta\rightarrow\infty$, traditional perturbation theory gives us: $$ n = \int_{0}^{\mu}g(\epsilon)\text{d}\epsilon + \frac{\pi^2}{6}\frac{g'(\mu)}{\beta^2}+O(\beta^{-4}) $$
Under the Sommerfeld Model in three dimensions, $g(\epsilon)\propto\sqrt\epsilon$, so everything is copacetic. However, in two dimensions, $g(\epsilon)\propto1$, so the terms with explicit $\beta$-dependence go to $0$ exactly.
The leading order terms of the perturbation in 2 dimensions are sub-polynomial. In particular, this integral can be used to estimate how $\mu$ (the electron chemical potential) varies from its ground-state value, $\epsilon_F$ (the Fermi Energy). In three dimensions, a linear approximation (in powers of $\mu-\epsilon_F$) for the integral term is sufficient, but in two dimensions, any polynomial expansion for the integral term implies that $\mu=\epsilon_F$ (i.e. the chemical potential is constant for all values of $\beta$).
However, the two-dimensional integral can be evaluated explicitly and the relationship $$ \epsilon_F=\mu+\frac{1}{\beta}\ln(1+e^{-\mu\beta})\approx\mu+\frac{e^{-\mu\beta}}{\beta} $$ is found, which clearly varies from the ground state only by a transcendentally small term.
A natural approach might be to attempt an asymptotic expansion that extracts the exponential behavior ahead of time, but it is not obvious to me how that could be done for the specific two dimensional case much less for the general perturbation expansion for generic $g(\epsilon)$.
Question: Is there any way to have anticipated and corrected for the failure of the power series expansion to find these transcendentally small terms? Are transcendentally small terms simply invisible to perturbation theory, or if you know they are coming (especially if they are leading order) can you capture them?