If $a_n$ is a strictly increasing unbounded sequence, does $\sum_n \frac{a_{n+1} - a_n}{a_n}$ diverge? So I've been thinking through some test cases. If $a_n = n$ then $\sum_n \frac{a_{n+1} - a_n}{a_n}$ is the harmonic series which diverges. And if $a_n = \sum_{k=1}^n 1/k$ then $\sum_n \frac{a_{n+1} - a_n}{a_n}$ diverges like $\sum_n 1/(n \log n)$. So that got me thinking, if $a_n$ is a strictly increasing unbounded sequence, does $\sum_n \frac{a_{n+1} - a_n}{a_n}$ necessarily diverge?

Yes! 
$$\sum_{k=1}^p \frac{a_{n+k+1} - a_{n+k}}{a_{n+k}}\geq\sum_{k=1}^p\frac{a_{n+k+1} - a_{n+k}}{a_{n+p+1}} =\frac{a_{n+p+1}-a_{n+1}}{a_{n+p+1}} = 1-\frac{a_{n+1}}{a_{n+p+1}}$$
Notice $\lim\limits_{p\to\infty}\frac{a_{n+1}}{a_{n+p+1}}=0$, exists $N$, $\forall p\gt N$, such that 
$$\sum_{k=1}^p \frac{a_{n+k+1} - a_{n+k}}{a_{n+k}}\gt \frac12$$
