You are right. The key to understanding the paradox is that although Achilles must complete an infinite number of tasks in order to catch up to the Tortoise, he can do so in a finite amount of time, since each successive task takes much less time than its predecessor (as you noted). Of course, today we understand how to add an infinite sequence of terms that converge to a finite quantity. But this wasn't well understood until millenia after Zeno -- and the logical foundations for doing so required Cauchy and Weierstrass in the nineteenth century. So we shouldn't be too hard on old Zeno.
By the way, you might find it amusing to consider some more recent Zeno-like puzzles, such as the "New Zeno" discussed by Stephen Yablo in the journal ANALYSIS, vol 60 (April 2000).