[Nonexistence of boundary between convergent and divergent series?](https://mathoverflow.net/questions/49415/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 $\sum\_i \frac{a\_i}{b\_i} < \infty$.
> 
> 
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On convergent series - in the spirit of Abel and Dini
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.


