The multipole expansion makes it easy to know the general orders of energy that each term brings with it, which is very usefull for perturbation theory as this makes it clear which terms can be regarded as perturbations and which terms can not. One can only truncate when you are sure that the states that get truncated will only alter the energy very slightly and such a precision in energy is not required. also the total probabolty of states needs to be normalizable. I think that such a trunctauation should be possible.