The renormalization group (RG) is known to provide information about
radiative corrections beyond the order in perturbation theory to which one has
calculated explicitly. We first demonstrate the effect of the renormalization
scheme used on these higher order effects determined by the RG. Particular
attention is payed to the relationship between bare and renormalized
quantities. Application of the method of characteristics to the RG equation to
determine higher order effects is discussed, and is used to examine the free
energy in thermal field theory, the relationship between the bare and
renormalized coupling and the effective potential in massless scalar
electrodynamics
