The generalized random phase approximation is used to investigate the effects of electron-electron interaction on the spin susceptibility of an electron gas. Within this approximation, the induced spin density is determined by the solution of an integral equation that describes exchange scattering between virtually excited particles. Solutions to this equation are obtained in simple cases. They indicate that the exchange scattering enhances the susceptibility over that predicted by the Ruderman-Kittel formula by factors which, for electron densities comparable to those found in good metals, are in the range 1 to 2. These results can be derived by summing certain classes of diagrams in many-body perturbation theory. The structure of these graphs shows that the random phase approximation considers the simplest sort of self-energy effect and the simplest type of scattering of virtually excited particles. (auth)