Paradojas Matemáticas - Northrop Eugene 4682u

Outward 'Bound

120

of all the of the infinite class in example (3) above. If we write 2

+ 4 + 8 + 16 + 32 + 64 + 128 + 256 + ... ,

it is evident at once that as we go on adding successive , the sum rapidly becomes larger and larger. Actually, it is not

12 + 4 + 8 + 16 + 32 r'IG.

74. The sum 2 + 4

+8+

16 + 32 + 64 +128 + 256·· ·increases without limit

enough to say simply "larger and larger." We must be more precise. Let us note that by going out far enough in the series, we can make the sum of all the up to that point exceed any finite number, however large. This fact is indicated graphically in Figure 74. For example, if someone names the finite number 1000, we can, by taking 9 , make the sum 1022. If he raises the bid to 1,000,000, we can make the

Paradoxes of the Infinite

121

sum 1,048,574 by taking 19 . If he cares to go as high as 1,000,000,000, we have only to take 29 to make the sum 1,073,741,822. No matter how large a finite number our imaginary adversary sees fit to choose, it is evident that we can always make our sum exceed his number by taking a sufficiently large finite number of . This is what the mathematician means when he says that "the sum of this infinite series is infinite." But now let us return to the problem of motion from one point to another. Suppose the distance from P to Q is 100 yards, and that we walk at the rate of 100 yards per minute. Then the time required for the first stage of the journeyhalf the distance from P to Q-is ~ minute; that for half the remaining distance, U minute; that for half the distance then remaining, Ys minute; that for half the distance then remaining,7l6 minute; and so on. In other words, the time in minutes required to go from P to Q is the sum of the infinite series 1111111 1 "2 + "4 + "8 + 16 + 32 + 64 + 128 + 256 + .... (Note that this is the sum of all the of the infinite class in example (4) above.) Is the sum of this infinite series infinite? As in our previous series, the sum does get larger and larger as we go on adding successive . But it is not true that we can make the sum exceed any large finite number which anyone cares to name. A glance at Figure 75 shows us intuitively that the sum approaches more and more nearly to 1, but never exceeds it. More precisely, if anyone names a finite number, however small, we can, by taking a sufficiently large number of , make the difference between our sum and 1 smaller than the named number. For example, if someone chooses the number

10~0'

make the sum differ from 1 by

we can, by taking 10 ,

10~4·

If he lowers the bid to

1,00~,000 ' we can make the sum differ from

1 by 1,04!,576

1 by taking 20 . If he cares to go as low as 1,000,000,000 '