TITLE: Prove $\sum^{\infty}_{n=1} \frac{a_{n+1}-a_{n}}{a_{n}}=\infty$ for an increasing sequence $a_n$ of positive integers
QUESTION [16 upvotes]: The $a_n$'s are integers, positive, and increasing: $0< a_1 < a_2 < \cdots$, the problem asks us to prove that:
$$
\sum^{\infty}_{n=1} \frac{a_{n+1}-a_{n}}{a_{n}}=\infty
$$ 
While I have checked this results for several series like $a_n = n$, $a_n = n^2$, $a_n = n^p$, or $a_n = p^n$ type stuff, I don't know how to prove this general result. A hint is appreciated. Thanks dudes!

REPLY [9 votes]: We have $$\sum_{i=1}^{n} \frac{a_{i+1}-{a_i}}{a_i} \geq \sum_{i=1}^{n}\int_{a_i}^{a_{i+1}}\frac{1}{x}\rm{d}x=\ln\left(\frac{a_{n+1}}{a_1} \right )$$
taking $n\rightarrow \infty$ we get the result.