On the Divergence of the Perturbation Series for the Excluded-Volume Problem in Polymers

For a lattice polymer on the cubic lattice and for a Gaussian polymer in three dimensional space, it is proved that the square of the expansion factor α is zero for negative real z. the measure of the strength of the intrachain interaction, when the polymer length becomes infinite under the condition that z is kept finite. This fact implies that the perturbation series for α²(z) is not convergent except at z=0.