Imagine a pie, infinitely sliced as shown.

The first slice is

If we add up all the slices, we will obviously get one whole pie! So, curiously, the answer to our infinite sum is actually 1:

This equation is amazing. Since the sum keeps increasing without end, one would expect it to^{1}⁄_{2}+^{1}⁄_{4}+^{1}⁄_{8}+^{1}⁄_{16}+^{1}⁄_{32}+ … = 1

Of course, many infinite sums

1 + 2 + 3 + 4 + … = ∞But this is not always true and, indeed, the result of an infinite sum is not always predictable.

^{1}⁄_{2}+^{1}⁄_{2}+^{1}⁄_{2}+^{1}⁄_{2}+ … = ∞

From our examples, it may seem that if the individual numbers

We can’t prove this (well-known) result here, but such infinite sums have been investigated by mathematicians for centuries, with the full fruit of their labours being realized only in the 19th century.^{1}⁄_{2}+^{1}⁄_{3}+^{1}⁄_{4}+^{1}⁄_{5}+ … = ∞

For our purposes, it suffices to have vindicated Achilles. As we saw, these two facts about Achilles cannot be denied:

To reach the doorway, Achilles must undertake an infinite sequence of movements.But it doesEach movement will occupy some of his time.

In terms of our example, that would follow only if this equation were correct:

But we saw that this is wrong; the correct sum is actually 1.^{1}⁄_{2}+^{1}⁄_{4}+^{1}⁄_{8}+^{1}⁄_{16}+^{1}⁄_{32}+ … = ∞

Indeed, in our example, Achilles stands one meter from the doorway and walks at one meter per second. So we would expect him to take

The question now is whether this resolves the paradox.

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