In mathematics, perturbation theory works typically by expanding unknown quantity in a power series in a small parameter.
However, in a perturbation problem beyond all orders, all coefficients of the perturbation expansion vanish and the difference between the function and the constant function 0 cannot be detected by a power series.
A simple example is understood by an attempt at trying to expand
All terms in a naïve Taylor expansion are identically zero.
-plane, and therefore the function is most appropriately modeled by a Laurent series -- a Taylor series has a zero radius of convergence.
Thus, if a physical problem possesses a solution of this nature, possibly in addition to an analytic part that may be modeled by a power series, the perturbative analysis fails to recover the singular part.
are considered to be "beyond all orders" of the standard perturbative power series.