Fifth-order corrected expressions for the fields of a radially polarized Laguerre-Gauss (R-TEM(n1)) laser beams are derived based on perturbative Lax series expansion. When the order of Laguerre polynomial is equal to zero, the corresponding beam reduces to the lowest-order radially polarized beam (R-TEM(01)). Simulation results show that the accuracy of the fifth-order correction for R-TEM(n1) depends not only on the diffraction angle of the beam as R-TEM(01) does, but also on the order of the beam.