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High temperature expansions for the susceptibility and the second correlationmoment of the classical N-vector model (also known as the O(N) symmetricHeisenberg classical spin model or the as the lattice O(N) nonlinear sigmamodel) on the square lattice are extended from order beta^{14} to beta^{21} forarbitrary N. For the second field derivative of the susceptibility the seriesexpansion is extended from order beta^{14} to beta^{17}. For -2 < N < 2, anumerical analysis of the series is performed in order to compare the criticalexponents gamma(N), nu(N) and Delta(N) to exact (though nonrigorous) formulasand to compute the "dimensionless four point coupling constant" g_r(N). For N >2, we present a study of the analiticity properties of chi, xi etc. in thecomplex beta-plane and describe a method to estimate the parameters whichcharacterize their low-temperature behaviors. We compare our series estimatesto the predictions of the perturbative renormalization group theory, to exact(but nonrigorous or conjectured) formulas and to the results of the 1/Nexpansion, always finding a good agreement.Comment: 31 pages, Latex; abridged version without figures, the printed  version contains 5 figure

Perturbative renormalization group, exact results, and high-temperature series to order 21 for theN-vector spin models on the square lattice