$\sum_{n=1}^\infty \frac{a_{n+1}-a_n}{a_{n+1}}$ diverges Let $(a_n)_{n\ge 1}$ is an increasing sequence of of positive numbers such that $a_n \to \infty$ as $n \to \infty $. Then $\sum_{n=1}^\infty \frac{a_{n+1}-a_n}{a_{n+1}}$ diverges.

Verify that the Cauchy criterion isn't satisfied: Since $(a_n)$ is monotonically increasing,
$$\sum_{n=k}^m \frac{a_{n+1}-a_n}{a_{n+1}} \geqslant \sum_{n=k}^m \frac{a_{n+1}-a_n}{a_{m+1}} = \frac{a_{m+1}-a_k}{a_{m+1}}.$$
