Hard sphere perturbation theory expressions for the thermodynamic properties and the infinite frequency elastic moduli of fluids interacting with steeply repulsive pair potentials with the analytical form phi(r) = epsilon(sigma/r)(n) where epsilon and sigma set the energy and distance scales, respectively, are tested against extensive molecular dynamics (MD) simulation data. The convergence of these expressions as a function of the softness parameter n(-1) is examined by comparing with virtually exact values obtained from MD simulations of fluids interacting with these potentials. The value of the parameter n in the simulations ranged from 18 to the unusually high value of 288. Perturbation theory reproduces the thermodynamic properties and the infinite frequency elastic moduli from simulation, within the MD statistical uncertainty for n greater than 36. The self-diffusion coefficient D and shear viscosity eta(s) were determined also and are found to be quite sensitive to the value of n in the range studied. The convergence towards the hard sphere value is nonlinear in n(-1) for D at high fluid densities. At high densities the shear stress autocorrelation function decays increasingly rapidly with time, and the associated shear stress relaxation time diminishes according to n(-1) in the hard sphere limit, as predicted by perturbation theory using the Barker-Henderson equivalent hard sphere diameter
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