Nonexistence of boundary between convergent and divergent series?
I'm hoping the following is true:
Suppose $a_i $ is a positive sequence and $\sum_i a_i < \infty.$ Then there exists a positive sequence $b_i$ s.t $\sum_i b_i < \infty$ and $\sum_i \frac{a_i}{b_i} < \infty$.
This is not correct. Take $a_n=1/n^2$.
Let us show that $b_n$ with required properties does not exist. Consider the set $$E=\{ n: b_n\geq 1/n\}.$$ As $\sum b_n<\infty$, we have $$\sum_E1/n<\infty.$$ Now on $N\backslash E$ we have $b_n<1/n,$ so $1/b_n>n$ and as $\sum a_n/b_n<\infty$, we conclude that $$\sum_{N\backslash E}1/n<\infty.$$ adding the last two inequalities we obtain a contradiction.
