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We give an analytical derivation of the mass gap of the O(N) sigma models andinvestigate a large-order behavior of the weak coupling asymptotic expansionfor the energy. For sufficiently large N the series is sign-oscillating, whichis expected from the large N solution of the sigma model. However, for N=3 andN=4 the series are sign positive.

From the mass gap in O(N) to the non-Borel-summability in O(3) and O(4) sigma-models