The effects of finite size on the thermodynamic properties of ferromagnetic clusters have been calculated on the basis of a self-consistently solved spin-wave spectrum for clusters of various sizes. The finite size leads to an effective power law for the temperature dependence of the magnetization (M approximately T^alpha; 3/2< alpha <3), and to a neutron-scattering cross section with a wavevector broadened discrete spin-wave spectrum very different from that of the bulk. Predictions of the implications of finite size are extended to the size range of nanoparticles. The effects should be readily observed in experiments.