Five hundred years before Christ, the Greek philosopher Zeno was born. He is mostly famous for these paradoxes:

"Achilles is located a hundred yards south of the turtle; both of them travel north, with Achilles going 10 times faster than the turtle. The arguments are numbered on the right.

While Achilles bridges the 100 yards between them, the turtle moves 10 yards north. (1)

While Achilles travels the next 10 yards, the turtle travels 1 yard. (2)

While Achilles travels the next 1 yard, the turtle travels 0.1 yard. (3)

While Achilles travels the next 0.1 yard, the turtle travels 0.01 yard. (4)

etc... (…)

The turtle makes smaller and smaller steps but is always ahead of Achilles!!!! "

"I travel at the fixed speed of 1 yard per second and I want to go 100 yards. Clearly I first have to cover the first 50 yards. Let's say I do. I now have to go 50 yards. I get through the first 25 yards and now my job is to go 25 yards. Since I can continue this reasoning forever, I will always have some small distance to cover! The distance will get smaller and smaller, but is always greater than zero. Hence, I will never get there and any motion is impossible!"

These similar paradoxes are very hard to believe and very hard to refute. If you cannot refute them, you have to accept the sad fact that motion is impossible, even though our senses tell us that motion does exist. That was Zeno's conclusion. It took mankind 2,000 more years and the study of series to really solve these paradoxes.

This is an advanced branch of mathematics that you will not see for another 6 or 8 years, but I'll take a chance anyway. A series is an infinite sum of terms. For instance,

1 + 1 + 1 + 1 + 1 + 1 + 1 + . . .

1 + 2 + 3 + 4 + 5 + 6 + 7 + . . .

1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 +1/7 + . . .

Mathematicians are interested in what happens to each of these sums as the number of terms goes to infinity. In the three previous cases, the series also goes to infinity (we say that the series does not converge). But, this is not always true! We are going to look at three other cases where we say that the series converge (to a finite limit):

0.9 + 0.09 + 0.009 + 0.0009 + 0.00009 + 0.000009 + . . .

We can write this sum as 0.999999… and then remember that we saw this last week! We had "proved" that this was actually equal to 1. Well, it is still true this week. Even though the partial sums increase forever, they are always smaller than 1. As the number of terms tends to infinity, the series tends to the finite number 1 (we also say that its limit is 1). Similarly,

1 + .1 + .01 + .001 + .0001 + .00001 +.000001 + ...

is always increasing as we keep adding infinitely many terms. Yet its limit is finite. Notice that the partial sums are all less than 1.2. The limit is exactly 10/9. This is the magic fact we need to solve Zeno's paradox of Achilles and the turtle.

Let us say that Achilles takes one minute to go the first 100 yards;

Achilles goes 100 yards, turtle goes 10 yards, 1 minute elapse, clock=1.0

Achilles goes 10 yards, turtle goes 1 yard, 0.1 minute elapse, clock=1.1

Achilles goes 1 yard, turtle goes 0.1 yard, 0.01 minute elapse, clock=1.11

Achilles goes 0.1 yard, turtle goes 0.01 yard, 0.001 minute elapse, clock=1.111

Achilles goes 0.01 yard, turtle goes 0.001 yard, 0.0001 minute elapse, clock=1.1111

etc... (…)

Altogether, my arguments cover 1.11111111...minutes. That time is not infinite, we just saw that it is finite and equal to 10/9 minutes. My arguments, even though each of them is correct and even though they are infinite in number, only look at a finite horizon. The reasoning is absolutely correct but only covers what happens up to time 10/9 minutes; it does not cover what happens after that!

Let us look at the series

1/2 + 1/4 + 1/8 + 1/16 + 1/32 + . . .

Let's compute the partial sums and try to find a pattern:

1/2 = 1/2

1/2 + 1/4 = 3/4

1/2 + 1/4 + 1/8 = 7/8

1/2 + 1/4 + 1/8 + 1/16 = 15/16

1/2 + 1/4 + 1/8 + 1/16 + 1/32 = 31/32

If you look carefully, you will notice the pattern. This noticing of patterns is what intelligence is all about; it is also what computers seriously lack, even though they can compute millions (or billions) of times faster than you. Do not look below my signature until you have guessed the pattern and therefore the limit of the series. Once you have, you can find the flaw in Zeno's second paradox!

 Jean-Claude

The series converges to 1. The flaw is the same as for Achilles.

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