We study the topology of all possible subsums of the generalized multigeometric series$$k_1f(x)+k_2f(x)+\dots +k_mf(x)+\dots + k_1f(x^n)+\dots +k_mf(x^n)+\dots ,$$
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f
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+
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f
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+
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+
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m
f
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k
1
f
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f
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,
where $$k_1, k_2, \dots , k_m$$
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,
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,
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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.