We present an application of Wertheim's thermodynamic perturbation theory (TPT1) to a simple coarse grained model made of flexibly bonded Lennard-Jones monomers. We use both the reference hypernetted chain (RHNC) and mean spherical approximation (MSA) integral equation theories to describe the properties of the reference fluid. The equation of state, the density dependence of the excess chemical potential, and the critical points of the liquid-vapor transition are compared with simulation results and good agreement is found. The RHNC version is somewhat more accurate, while the MSA version has the advantage of being almost analytic. We analyze the scaling behavior of the critical point of chain fluids according to TPT1 and find it to reproduce the mean field exponents: The critical monomer density is predicted to vanish as n(-1/2) upon increasing the chain length n while the critical temperature is predicted to reach an asymptotic finite temperature that is attained as n(-1/2). The predicted asymptotic finite critical temperature obtained from the RHNC and MSA versions of TPT1 is found to be in good agreement with the Theta point of our polymer model as obtained from the temperature dependence of the single chain conformations.