Let $\displaystyle \{a_n\}$ be a strictly increasing sequence of positive integers.

Can the series

$\displaystyle \sum^{\infty}_{n=1}\left(1-\frac{a_n}{a_{n+1}}\right)=\left(1-\frac{a_1}{a_2}\right)+\left(1-\frac{a_2}{a_3}\right)+\ldots$

ever converge?