  We study the O(N) linear sigma model in 1+1 dimensions. We use the 2PI
formalism of Cornwall, Jackiw and Tomboulis in order to evaluate the effective
potential at finite temperature. At next-to-leading order in a 1/N expansion
one has to include the sums over "necklace" and generalized "sunset" diagrams.
  We find that - in contrast to the Hartree approximation - there is no
spontaneous symmetry breaking in this approximation, as to be expected for the
exact theory. The effective potential becomes convex throughout for all
parameter sets which include N=4,10,100, couplings lambda=0.1 and 0.5, and
temperatures between 0.2 and 1. The Green's functions obtained by solving the
Schwinger-Dyson equations are enhanced in the infrared region. We also compare
the effective potential as function of the external field phi with those
obtained in various other approximations.
