The asymptotic behavior of the atomic wave functions is used to show that high-order multipole moment matrix elements have a sharp peak when the excitation energy is equal to twice the ionization energy. This makes it possible to deduce the correct forms of the weighted sums involving these matrix elements. In this paper we demonstrate in the case of hydrogen and helium how these correct forms can be used to obtain accurate polarizabilities alpha(L) and dispersion coefficients C-2n Of arbitrary high L and n. (C) 1997 American Institute of Physics.