1⁄2 + 1⁄4 + 1⁄8 + 1⁄16 + 1⁄32 + ... = 1This equation is amazing. Since the sum keeps increasing without end, one would expect it to blow up to infinity. But the pie shows otherwise and here’s another nice example: Here too the sum increases endlessly but the result is obviously correct. So, amazingly, an infinity of positive numbers can sum to a finite quantity.
1 + 2 + 3 + 4 + ... = infinityBut 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 + ... = infinity
1⁄2 + 1⁄3 + 1⁄4 + 1⁄5 + ... = infinityWe 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 19th century Europe.
To reach the doorway, Achilles must undertake an infinite sequence of movements.But it does not follow that, to complete the undertaking, Achilles requires an infinite amount of time.
Each movement will occupy some of his time.
1⁄2 + 1⁄4 + 1⁄8 + 1⁄16 + 1⁄32 + ... = infinityBut we saw that this is wrong; the correct sum is actually 1.