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?

Suppose that $\frac{a_{n+1}-a_n}{a_n}=b_n$, and $\sum\limits_{n=1}^\infty b_n<\infty$.
Then
$$1+b_n=\frac{a_{n+1}}{a_n}$$
hence,
$$a_{n+1}=\prod_{k=1}^n (1+b_n)$$
Transforming the product into the exp of a sum and taking limits,
$$\lim_{n\to\infty} a_n=\exp\left(\sum_{n=1}^\infty\ln(1+b_n)\right)<\exp\sum_{n=1}^\infty b_n<\infty$$
So if $a_n$ is not bounded, your sum must diverge.
