@bender1978asymptotic]. Furthermore, there are countless examples where *perturbative expansions* of physical quantities admit asymptotic expansions of this kind [@bender1969anharmonic; @le2012large; @dunne2012resurgence].

The restriction to this specific class of power series is inspired by the work of Bender [@bender1975asymptotic]. In this work the asymptotic behaviour of the composition of a power series, which has mildly growing coefficients, with a power series, which has rapidly growing coefficients, is analyzed. Bender’s results are extended into a complete algebraic framework. This is achieved making heavy use of generating functions in the spirit of the analytic combinatorics or ‘generatingfunctionology’ approach [@flajolet2009analytic; @wilf2013generatingfunctionology]. The key step in this direction is to interpret the *coefficients of the asymptotic expansion as another power series*.

These structures bear many resemblances to the theory of resurgence, which was established by Jean Ecalle [@ecalle1981fonctions]. Resurgence assigns a special role to power series which diverge factorially, as they offer themselves to be Borel transformed. Jean Ecalle’s theory can be used to assign a unique function to a factorially divergent series. This function could be interpreted as the series’ generating function. Moreover, resurgence provides a promising approach to cope with divergent perturbative expansions in physics. Its application to these problems is an active 