The question of whether the known virial coefficients are enough to determinethe packing fraction $\eta_\infty$ at which the fluid equation of state of ahard-sphere fluid diverges is addressed. It is found that the informationderived from the direct Pad\'e approximants to the compressibility factorconstructed with the virial coefficients is inconclusive. An alternativeapproach is proposed which makes use of the same virial coefficients and of theequation of state in a form where the packing fraction is explicitly given as afunction of the pressure. The results of this approach both for hard-disk andhard-sphere fluids, which can straightforwardly accommodate higher virialcoefficients when available, lends support to the conjecture that $\eta_\infty$is equal to the maximum packing fraction corresponding to an orderedcrystalline structure.
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