# Interactive Real Analysis

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## 4.1. Series and Convergence

### Example 4.1.1: Zeno Paradox (Achilles and the Tortoise)

Achilles is racing against a tortoise. Achilles can run 10 meters per second, the tortoise only 5 meter per second. The track is 100 meters long. Achilles, being a fair sportsman, gives the tortoise 10 meter advantage. Who will win ?
Let us look at the difference between Achilles and the tortoise:
 Time Difference t = 0 10 meters t = 1 5 = 10 / 2 meters t = 1 + 1/2 2.5 = 10 / 4 meters t = 1 + 1/2 + 1/4 1.25 = 10 / 8 meters t = 1 + 1/2 + 1/4 + 1/8 0.625 = 10 / 16 meters
and so on. In general we have:
 Time Difference t = 1 + 1 / 2 + 1 / 2 2 + 1 / 2 3 + ... + 1 / 2 n 10 / 2 n meters
Now we want to take the limit as n goes to infinity to find out when the distance between Achilles and the tortoise is zero. But that involves adding infinitely many numbers in the above expression for the time, and we don't know how to do that. However, if we define
S n = 1 + 1 / 2 + 1 / 2 2 + 1 / 2 3 + ... + 1 / 2 n
then, dividing by 2 and subtracting the two expressions:
S n - 1/2 S n = 1 - 1 / 2 n+1
or equivalently, solving for S n:
S n = 2 ( 1 - 1 / 2 n+1)
But now S n is a simple sequence, for which we know how to take limits. In fact, from the last expression it is clear that
lim S n = 2
as n approaches infinity. Hence, we have - mathematically correct - computed that Achilles reaches the tortoise after exactly 2 seconds, and then, of course passes it and wins the race.

A much simpler calculation not involving infinitely many numbers gives the same result:

• Achilles runs 10 meters per second, so he covers 20 meters in 2 seconds
• The tortoise runs 5 meters per second, and has an advantage of 10 meters. Therefore, it also reaches the 20 meter mark after 2 seconds
• Therefore, both are even after 2 seconds
Of course, Achilles will finish the race after 10 seconds, while the tortoise needs 18 seconds to finish, and Achilles will clearly win.

The problem with Zeno's paradox is that Zeno was uncomfortable with adding infinitely many numbers. In fact, his basic argument was: if you add infinitely many numbers, then - no matter what those numbers are - you must get infinity. If that was true, it would take Achilles infinitely long to reach the tortoise, and he would loose the race. However, reducing the infinite addition to the limit of a sequence, we have seen that this argument is false.

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