A review is made of the calculation of multipole polarizabilities in three dimensions, for hydrogen-like ions with nuclear charge Z, in which the solutions of the inhomogeneous equations in perturbation theory are used. The procedure is extended to arbitrary dimension D, and an explicit expression for exact multipole polarizabilities is obtained. A calculation of an approximate ground state energy using 1/D expansions is made, leading to a classic harmonic oscillator problem. The possibility to estimate the dipole polarizability in the large D limit from the obtained expressions is used. A comparison with the exact result yields quantitative agreement in the coefficient as well as in the D dependence, when using a physically motivated form of the perturbation. Inequalities for oscillator strengths, previously used for estimating dipole polarizabilities in three dimensions, are generalized to D dimensions, and expressions for the dipole polarizability in the large D limit are obtained. The exact results, the dimensional scaling calculations, and the expressions obtained from inequalities are compared and evaluated. It is shown that the exact first order correction to the unperturbed wave function reduces to one term in the sum over states expression. The asymptotic result for the dipole polarizabilities is, in atomic units, alpha2 = (64Z4)-1 D6. in Dimensional