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bijection between convergent and divergent series

Source: miklos schweitzer 1996 q5

October 15, 2021
real analysis

Problem Statement

Let K and D be the set of convergent and divergent series of positive terms respectively. Does there exist a bijection between K and D such that for all ∑an,∑bn∈K\sum a_n,\sum b_n\in K and ∑an′,∑bn′∈D\sum a_n',\sum b_n'\in D , anbn→0  ⟺  an′bn′→0\frac{a_n}{b_n}\to 0\iff \frac{a_n'}{b_n'}\to 0 ? Under the bijection, ∑an↔∑an′\sum a_n\leftrightarrow\sum a_n' and ∑bn↔∑bn′\sum b_n\leftrightarrow\sum b_n'.
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