The nuclear magnetic resonance of paramagnetic solids is usually characterized by the presence of large chemical shifts and shift anisotropies due to hyperfine interactions. Frequently the resulting spectra cover a frequency range of several megahertz, which is greater than the bandwidth of commercially available radio-frequency (RF) probes, making it impossible to acquire the whole spectrum in a single experiment. In these cases it common to record a series of spectra, in which the probe is tuned to a different frequency for each, and then sum the results to give the ``true'' spectrum. While this method is very widely used on static samples, the application of frequency stepping under magic-angle spinning (MAS) is less common, owing to the increased complexity of the spin dynamics when describing the interplay of the RF irradiation with the mechanical rotation of the shift tensor. In this paper, we present a theoretical description, based on the jolting frame formalism of Caravatti et al. [J. Magn. Reson. 55, 88 (1983)], for describing the spin dynamics of a powder sample under MAS when subjected to a selective pulse of low RF-field amplitude. The formalism is used to describe the frequency stepping method under MAS, and under what circumstances the true spectrum is reproduced. We also present an experimental validation of the methodology under ultra-fast MAS with the paramagnetic materials LiMnPO4 and TbCsDPA. (C) 2013 American Institute of Physics. [http://dx.doi.org/10.1063/1.4795001]