11 pages, 6 figures.- Submitted to Euro. Phys. J. B.Here, in order to study stochastic resonance (SR) in a double-well potential when the noise source has a spectral density of the form f^{-κ} with varying κ, we have extended a procedure, introduced by Kaulakys et al (Phys. Rev. E 70, 020101 (2004)). In order to have an analytical understanding of the results, we have obtained an effective Markovian approximation, that allows us to make a systematic study of the effect of such kind of noises on the SR phenomenon. The comparison of numerical and analytical results shows an excellent qualitative agreement indicating that the effective Markovian approximation is able to correctly describe the general trends.MAF would like to thank the support of CONICET, Argentina, to the Santa Fe Institute for its support and hospitality, and to M. Ballard for fruitful discussions related to the one-dimension FPE description. HSW wants to thank the European Commission for the award of a Marie Curie Chair.Peer reviewed