We consider weakly singular perturbations λ¦x¦−ν(0<ν<2) of an even restoring potential. We compute the matrix elements of the perturbation together with the additional point potential associated with the perturbation. It is shown that even for unperturbed wave functions, the matrix elements exist when 0 < ν < 3/2. The series for the Rayleigh-Schrodinger coefficients converge in all orders for the same interval in ν, regardless of the form of the restoring potential. For odd states, the matrix elements of the perturbation exist when 0 < ν < 3, while estimates for the Rayleigh-Schrodinger coefficients give the boundary ν = 2.