If a positive increasing sequence tends to infinity, then $\sum_{n=2}^\infty \frac{a_n-a_{n-1}}{a_n}=\infty$
If the positive series $\sum a_n$ diverges and $s_n=\sum\limits_{k\leqslant n}a_k$ then $\sum \frac{a_n}{s_n}$ diverges as well