  The meaning of the inflationary slow-roll approximation is formalised.
Comparisons are made between an approach based on the Hamilton-Jacobi
equations, governing the evolution of the Hubble parameter, and the usual
scenario based on the evolution of the potential energy density. The vital role
of the inflationary attractor solution is emphasised, and some of its
properties described. We propose a new measure of inflation, based upon
contraction of the comoving Hubble length as opposed to the usual e-foldings of
physical expansion, and derive relevant formulae. We introduce an infinite
hierarchy of slow-roll parameters, and show that only a finite number of them
are required to produce results to a given order. The extension of the
slow-roll approximation into an analytic slow-roll expansion, converging on the
exact solution, is provided. Its role in calculations of inflationary dynamics
is discussed. We explore rational-approximants as a method of extending the
range of convergence of the slow-roll expansion up to, and beyond, the end of
inflation.
