We consider the thermodynamics of a uniformly charged polyelectrolyte with harmonic bonds. For such a system there is at high temperatures an approximate scaling of global properties like the end-to-end distance and the interaction energy with the chain length divided by the temperature. This scaling is broken at low temperatures by the ultraviolet divergence of the Coulomb potential. By introducing a renormalization of the strength of the nearest-neighbor interaction the scaling is restored, making possible an efficient blocking method for emulating very large polyelectrolytes using small systems. The high temperature behavior is well reproduced by the analytical high-T expansions even for fairly low temperatures and system sizes. In addition, results from low-T expansions, where the coefficients have been computed numerically, are presented. These results approximate well the corresponding Monte Carlo results at realistic temperatures. A corresponding analysis of screened chains is performed. The situation here is complicated by the appearance of an additional parameter, the screening length. A window is found in parameter space, where scaling holds for conformational as well as thermodynamical properties. This window corresponds to situations where the range of the potential interpolates between the bond length and the size of the chain. This scaling behavior, which is verified by Monte Carlo results, is, for the end-to-end distance, consistent with Flory scaling. Also for the screened chain a blocking approach can be devised, that performs well for low temperatures, whereas the low-T expansion is inaccurate at realistic temperatures.