We show that a series of recurrent inequalities derived in N = 3 have the same formal expressions in any dimension N ≥ 2. They are derived from the multipole sum rules, and provide us with upper bounds for the moments of the ground-state density depending only on the lowest multipole excitation energy. These bounds are transformed into approximate recurrent relations by means of an empirical correction factor. The 1/r potential and the harmonic oscillator play a key role in establishing this factor, which is exact for these two potentials by construction. For a large class of potentials, we show that this factor tends to 1 as N → ∞. In such cases, at the large-N limit, the lowest state for each multipole excitation exhausts the sum rule. It thus acquires the characteristics of the one-phonon excitation typical of the harmonic oscillator.