I had this question where it was given $a_n > 0$ and $\sum a_n$ convergent and I was to find out as well as prove that how does $$\sum a_n/r_n$$ behave where $$r_n = \sum_{m = n}^{\infty} a_m$$.
I attempted a method like this i think we can conclude from above given definition of $r_n$ that $r_n > r_{n + 1}$ and rewriting $a_n$ in $a_n/r_n$ as $(r_n - r_{n + 1})/r_n = 1 - \frac {r_{n + 1}}{r_n}$, now since this term is the general term of $\sum a_n/r_n$ and $$1 - \frac {r_{n + 1}}{r_n } \not= 0$$ so is it sufficient to conclude that $\sum a_n/r_n$ is divergent ?
I just couldn't assure myself completely with this method so I came here
Hint: prove that $$\dfrac{a_m}{r_m}+\cdots+\dfrac{a_n}{r_n}>1-\dfrac{r_n}{r_m}$$ if $m<n$ and use Cauchy criterion to conclude that $\sum \limits\dfrac{a_n}{r_n}$ diverges.