How was Zeno's paradox solved using the limits of infinite series?
It is often claimed that Zeno's paradoxes of motion were "resolved by" the infinitesimal calculus, but I don't really think this claim stands up to a closer investigations. It depends on very specific notion of what it would mean to "resolve" the paradox, and it looks to me that this notion can only be arrived at if one deliberately sets out to reformulate the paradox into a problem that
- Is somewhat related to the classical description of the paradox, and
- feels non-trivial enough that it's a problem worth solving, yet
- is simple enough that contemporary mathematics has a solution to it.
In this way the original "therefore Achilles never overtakes the tortoise!" becomes tacitly transformed into "WHEN does Achilles overtake the tortiose?" and once we get a numerical answer to the latter, the popularizers will claim that the original contradiction has been resolved.
Actually, the first step in resolving the paradox must be to understand why it was supposed to be paradoxical in the first place. Zeno's paradoxes rely on an intuitive conviction that
It is impossible for infinitely many non-overlapping intervals of time to all take place within a finite interval of time.
Only if we accept this claim as true does a paradox arise. But why should we accept that as true? If we reject it, then the paradox disappears -- and we don't need any infinitesimal calculus, or any concept of limit, to deny this premise. On the contrary, it must be up to anyone who wants us to accept it to provide some kind of argument -- and then he has the trouble of explaining why his pet principle doesn't conflict with reality, as demonstrated by Zeno!
Part of the story here is that the historical record is too fragmentary for us to know what the historical Zeno actually intended with the paradoxes. None of his own writings survive, and all we have to go by is other classical authors, who discussed the paradoxes for their own purposes, and just gave credit to Zeno for coming up with the example.
On this sparse record, it is entirely possible that Zeno never intended to convince anyone that motion is impossible. Instead, it is easy to imagine that, for example, one of Zeno's pupils may have tried to use the indented principle above in an argument, and then the master immediately shot that down by using the "paradox" as a vivid practical example of how there can be infinitely many instants or intervals in a finite period of time.
If the subject walks at a constant speed, you don't need calculus to calculate exactly when he will arrive at his destination. Just use the simple speed-equals-distance-over-time formula. This formula was unknown to the ancient Greek philosophers. They had no notion of actually measuring speed. It wouldn't be for another thousand years after Zeno's time that Galileo would formulate:
$$s=\frac d{t}$$
Today, we know that in going from point $A$ to another point $B$, we will pass through infinitely many other points along the way. If we associate an event with our arrival at each of these points, then infinitely many events will have occurred in the interval. The modern mind has no trouble with this notion.