SCHWEITZER 1996 Problem 5

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 $\sum a_n,\sum b_n\in K$ and $\sum a_n',\sum b_n'\in D$ , $\frac{a_n}{b_n}\to 0\iff \frac{a_n'}{b_n'}\to 0$ ? Under the bijection, $\sum a_n\leftrightarrow\sum a_n'$ and $\sum b_n\leftrightarrow\sum b_n'$ .