Rudin PMA p.63
$\sum 1/n$ diverges, but $\sum 1/{n^2}$ converges. Likewise, $\sum \frac 1{n \log n}$ diverges, but $\sum \frac 1{n (\log n)^2}$ converges.
This might lead us to the conjecture that there is a limiting situation of some sort "boundary".
However, the conjecture is false. The book says one can check this by showing following two theorems.
Theorem 1;
If $a_n >0$, $s_n=a_1 + \cdots + a_n$ and $\sum a_n$ diverges, then $\sum a_n / s_n$ diverges.
Theorem 2;
If $a_n>0$, $r_n=\sum_{m=n}^\infty a_m$ and $\sum a_n$ converges, then $\sum a_n / \sqrt r_n$ converges.
I have proved those two theorems, but still don't understand how are these two theorems related to "boundary conjecture"..