A recently proposed exact power expansion for the dynamic (frequency-dependent) multipole polarizabilities of a ground-state hydrogen atom has been improved keeping its simple polynomial structure substantially unaffected and enlarging its range of convergence. The improvement is performed by adding unexpanded frequency terms taken from the formal expression of the polarizability as a sum-over-states and subtracting their corresponding power expansions. The enlarged range of convergence obtained in this way covers all real frequencies of physical importance and the related imaginary frequencies.