> Clearly, not all infinite sequences can be summed. So e.g., 1, -1, 1, -1, … has no sum.
Your geometric series is not a summation of the steps or positions, but rather the time required to complete each step. Therefore your example is characterized by an identical geometric series to the model I used in my previous comment.
More generally, Zeno’s paradox can be succinctly resolved by citing the monotone convergence theorem. Every bounded, monotonically decreasing function converges. The time required to complete the infinite series of half steps converges, because (again, with the definition of a metric) the time required to complete each individual step decreases commensurate with the change in distance.
>Therefore your example is characterized by an identical geometric series to the model I used in my previous comment.
I am not sure what you mean here. You can calculate the sum of the time series, but you can't calculate Achilles' final position, which is the question at issue. The question remains: if it's possible in general to traverse an infinite sequence of steps in space, why is it not possible to traverse the one that I specified? "Solving" Zeno's paradox by admitting the possibility of traversing an infinite series of points in space or time seems to give rise to paradoxes just as deep as the originals.
> The time required to complete the infinite series of half steps converges, because (again, with the definition of a metric) the time required to complete each individual step decreases commensurate with the change in distance.
Yes, that was Aristotle's observation and a key part of his proposed solution to the paradox. The problem is that this explains why it's possible to sum the series, not why it's possible to traverse it. You seem to be taking the position that any series that cannot be summed cannot be traversed. But why should that be so?
Thinking about this a bit more, I think what I'm trying to say is that Zeno's paradox is more about supertasks than it is about the problem of computing the sum of an infinite series. There's a nice summary article that I found here:
Your geometric series is not a summation of the steps or positions, but rather the time required to complete each step. Therefore your example is characterized by an identical geometric series to the model I used in my previous comment.
More generally, Zeno’s paradox can be succinctly resolved by citing the monotone convergence theorem. Every bounded, monotonically decreasing function converges. The time required to complete the infinite series of half steps converges, because (again, with the definition of a metric) the time required to complete each individual step decreases commensurate with the change in distance.