Let $\sum_i a_i$ be a convergent sum with positive $a_i$. Does $\sum_i \frac{a_i}{a_i+a_{i+1}+a_{i+2}+\cdots}$ always diverge?
If $\sum a_n$ converges and $b_n=\sum\limits_{k=n}^{\infty}a_k $, prove that $\sum \frac{a_n}{b_n}$ diverges