2

and by writing the rest as:

limS
n→ ∞n

=1

+r+r

2

+r

3

+r

4

+r

5+L+r

n+L

Exercise 3: If ris a number such that 0

<r<1, what do you suppose is the value of

lim
n→ ∞

1

-rn+1
1-r

=

Hint: Try first calculating lim
n→∞

1

-1?

2

? ? ?
1- 1

2

n+1

=

Part 2: Zeno's Paradox

This supposedly is what Zeno said:

"Achilles cannot overtake a fleeing tortoise because in the interval of time that he
takes to get where the tortoise was, it can move away. But even if it should wait for
him, Achilles must first reach the halfway mark between them and he cannot do this
unless he first reaches the halfway mark to that mark, and so on indefinitely.
Against such an infinite conceptual regression, he cannot even make a start, and so
motion is impossible."

There actually are two different paradoxes contained in Zeno's statement. We will

separate them, calling the first the Achilles Paradoxand the second the Dichotomy

Paradox.

1.

Achilles Paradox: Achilles cannot overtake a fleeing tortoise because in the

interval of time that he takes to get where the tortoise was, it can move

away.

2.

Dichotomy Paradox: There is no motion because that which is moved must

arrive at the middle before it arrives at the end.

We are going to discuss the Achilles Paradox.

Suppose for sake of simplicity that although they start the race at the same time, the
tortoise (T) starts one foot ahead of Achilles (A). Also, assume that the tortoise moves
one-half foot, then one-quarter foot, then one-eighth foot, then one-sixteenth foot, and so