To cross a room you must first cover half of it, then half of what is left, and so on without end. Zeno concludes that you never arrive. The steps form a geometric series, and the lab computes its partial sums exactly: after ten halves the sum is 1023/1024, 1/1024 remains, and the limit is 1. The premise to deny is that infinitely many positive times must add to infinity.
Core ideas
The paradox is an argument about adding up, so the lesson computes the sums it is about.
Each step is half the one before. The first covers 1/2 of the room, the second 1/4, the third 1/8. A list in which each term is a fixed multiple of the last, added up, is a geometric series, with first term a = 1/2 and ratio r = 1/2.
The sum of the first n steps is a fraction, computed exactly. The lab prints it, and prints what remains of the distance. The pattern in those fractions is what the limit is.
The limit is 1 because the remainder, 1/2^n, falls below any fraction you name once n is large enough. The sums never reach 1 at a finite step, and they do not need to.
The material
The argument, the sums that answer it, and what the answer leaves alone.
Cross a room of length one. Before you can get all the way across you must get halfway. Before the rest, you must cover half of what remains, a quarter. Then an eighth, and the halving never stops, so the crossing has infinitely many parts. Zeno's argument is that a task with infinitely many parts cannot be finished.
The argument is valid, and the first two premises are as plausible as a description of walking can be. The third is the one with a hidden claim in it: it says that adding enough positive amounts always passes any size. That is true of equal steps, as the next lesson shows. Whether it is true of steps that shrink is a matter of arithmetic.
The sum of the first n halves is 1 − 1/2^n. After one step it is 1/2; after two, 3/4; after three, 7/8. The lab prints these as exact fractions. With the first preset, ten halves, it reports the sum 1023/1024 and the remainder 1/1024, which is the distance still to go. The limit tile shows 1.
What does it mean to say the infinite sum equals one, when no step of the lab adds infinitely many numbers? It means this: whatever small fraction you name, the remainder falls below it from some step on. Name one thousandth. After nine steps the remainder is 1/512, too large; after ten it is 1/1024, smaller than one thousandth, and it stays smaller for every step after. Name one millionth and the twentieth step does it. The third preset shows twenty steps: the remainder is 1/1048576. There is no number but 1 that the sums approach in this way, and that is all that “the limit is 1” says.
A walker who covers two thirds of what is left at each step has a = 2/3 and r = 1/3. After five steps the sum is 242/243 and the remainder is 1/243. The limit is again 1. The second preset is this walk, and a ratio of 1/3 shrinks the remainder faster than a ratio of 1/2.
So the third premise is false for these steps. A sum of infinitely many positive times is not always infinite: when the times shrink in a fixed proportion the total is finite, and here it is the whole crossing time. The argument went wrong in saying that infinitely many steps take infinitely long. They take exactly as long as the crossing, because the steps are the crossing.
The answer has limits and they should be stated. The arithmetic settles the sum, and the sum was the premise at fault. It does not settle whether space is made of points or can be divided without end, or whether a physical body can perform a task for every one of an unending list. Those are questions about the world and not about the series. The next lesson takes the paradox in which the same series appears with the speeds stated, and the lesson after that asks what the series cannot give.
Interactive
n terms added as fractions, against the limit when there is one
Method
Four moves, in this order, for any list in which each term is a fixed multiple of the last.
Write the first two steps. The first is a; the second divided by the first is r. For the halves, a = 1/2 and r = 1/2.
The sum of the first n terms is a·(1 − r^n) / (1 − r). For the halves it simplifies to 1 − 1/2^n.
The limit less the partial sum is what is left. For the halves it is 1/2^n, which halves with each new step.
If the ratio is between −1 and 1, it does, and the limit is a / (1 − r). If not, the sums do not settle and there is no limit.
For the dichotomy, exit one: the premise that infinitely many positive times must add to infinity is denied, and the series is the refutation. The price is a habit about adding up, and nothing about walking.
The room has length 1. The walker covers half, then half of what is left, ten times.
Every step leaves half the previous remainder, so the remainder after n steps is 1/2^n. It is under one thousandth from the tenth step on and under one millionth from the twentieth. The sums approach 1 and no other number.
Given the first term and ratio of a geometric series, you should be able to compute the partial sum after a stated number of terms as an exact fraction, state the remainder, find the number of steps after which it is below a given fraction, and state the limit or say that there is none.
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