  We study the topology of all possible subsums of the generalized
multigeometric series $k_1f(x)+k_2f(x)+\dots+k_mf(x)+\dots +
k_1f(x^n)+\dots+k_mf(x^n)+\dots,$ where $k_1, k_2, \dots, k_m$ are fixed
positive real numbers and $f$ runs along a certain class of non-negative
functions on the unit interval. We detect particular regions of this interval
for which this achievement set is, respectively, a compact interval, a Cantor
set and a Cantorval.
