Clearly, not all infinite sequences can be summed. So e.g., 1, -1, 1, -1, … has no sum.
Now suppose that Achilles takes alternate forward and backward steps a (countably) infinite number of times. The first step takes one second, the second step takes half a second, and so on. (Each step covers the same distance.) Where does he end up after 2 seconds?
There’s no sensible answer to that question. Does that mean that Achilles can’t in fact traverse that particular sequence of steps? But then, why should he be unable to traverse a particular infinite sequence of steps merely because its sum is undefined? After all, the result of each individual step is perfectly well defined. If it’s possible in general to traverse infinite sequences, what stops him traversing that one?
To me, this just seems like Zeno’s paradox all over again. The mathematical treatment is more sophisticated, but the underlying paradox remains.
Zeno himself probably wouldn’t have distinguished carefully between summing an infinite sequence and spatially or temporally traversing it, since both notions would have seemed equally absurd from his point of view. Modern mathematics has shown us that the former isn’t in fact absurd. But Zeno’s paradoxes are arguably about the latter.