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?

Question

Let $\sum_i a_i$ be a convergent sum, with all $a_i$ positive. Let $s_n=\sum_{i=n}^{\infty}a_i$.

Does $\sum_i \frac{a_i}{s_i}$ always diverge?

I've tried a few examples such as $a_i= r^i$ (geometric series) and $a_i=1/i^2$ and it seems to always diverge.