The fact that the geomagnetic potential, as well as magnetic induction and its gradient, can be represented with an arbitrary accuracy in the scope of multipole models has been justified programmatically and analytically. The obtained expressions are brief and can be utilized in analytical and computer-based studies using methods of computer algebra and numerical computations. Previously unknown analytical expressions for the components of the first seven multipole tensors, expressed in terms of the Gaussian coefficients, have been obtained for applied problems of space dynamics. An algorithm that makes it possible to construct analytical expressions for the arbitrary-order multipole tensor and magnetic induction vector components in any finite approximation has been developed.