Let $\sum a_n$ be a convergent series of positive numbers and set $r_n=\sum_{k=n}^\infty a_n$. Prove that the series $\sum \frac{a_n}{r_n}$ diverges.   Solution I have tried  By definition, $r_n=a_n+r_{n+1}$. Hence,  $\frac{r_{n+1}}{r_n}=1-\frac{a_n}{r_n}$,  $\frac{r_{n+2}}{r_n}=\big(1-\frac{a_n}{r_n}\big)\big(1-\frac{a_{n+1}}{r_{n+1}}\big)&gt;1-\frac{a_n}{r_n}-\frac{a_{n+1}}{r_{n+1}}$.  By induction we have  $\frac{r_{n+p}}{r_n}&gt;1-\frac{a_n}{r_n}-\frac{a_{n+1}}{r_{n+1}}- \dots \frac{a_{n+p}}{r_{n+p}}$.  As the series $\sum a_n$ is conv, we have that $r_{n+p} \to 0$ as $p \to \infty$.  Now what? I am stuck here.
Let $\displaystyle \sum a_n$ be  convergent series of positive terms and set  $\displaystyle b_n=\sum_{k=n}^{\infty}a_k$ , then prove that $\displaystyle\sum \frac{a_n}{b_n}$ diverges.   I could see that $\{b_n\}$ is monotonically decreasing sequence converging to $0$ and I can write $\displaystyle\sum \frac{a_n}{b_n}=\sum\frac{b_n-b_{n+1}}{b_n}$, how shall I proceed further?
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