The low-frequency magnetoelectronic properties of single-layer graphene in a periodic magnetic field are studied by the gradient approximation within the Peierls tight-binding model. A modulated magnetic field along the zigzag direction is chosen for a detailed investigation of optical absorption spectra. The electric polarization is taken to be perpendicular to the modulated direction. The absorption spectra exhibit many prominent and inconspicuous asymmetric peaks. They result from the transitions of the original and extra band-edge states, respectively. The prominent peaks are discussed in detail and they obey a simple optical selection rule, Delta n=vertical bar n(v)-n(c)vertical bar=1, similar to that in a uniform magnetic field. Such a rule can be ascribed to certain characteristics of the wave functions. The energy of the prominent peaks strongly depends on the strength, while it exhibits little dependence on the period. The abovementioned results can be verified by optical measurements.