Page 75 - Foundations Of Differential Calculus

58 3. On the Infinite and the Infinitely Small
102. From the summation of infinite series we can gather many results
that both further illustrate this theory of the infinite and also aid in an-
swering doubts that frequently arise in this material. In the first place, if
the series has equal terms, such as

1+1+1+1+1+ ··· ,
which is continued to infinity, there is no doubt that the sum of all of these
terms is greater than any assignable number. For this reason it must be
infinite. We confirm this by considering its origin in the expansion of the
fraction
1 2 3
=1 + x + x + x + ··· .
1 − x
If we let x = 1, then
1
= 1+1+1+1+ ··· ,
1 − 1
so that the sum is equal to

1 1
= = ∞.
1 − 1 0

103. Although there can be no doubt that when the same finite number
is added an infinite number of times the sum should be infinite, still, the
general infinite series that originates from the fraction
1 2 3 4 5
=1 + x + x + x + x + x + ···
1 − x
seems to labor under most serious difficulties. If for x we successively sub-
stitute the numbers 1, 2, 3, 4,..., we obtain the following series with their
sums:
1
A. 1+1+1+1+1+ ··· = 1−1 = ∞,
1
B. 1+2+4+8+16+ ··· = 1−2 = −1,
1 1
C. 1+3+9+27+81+ ··· = 1−3 = − ,
2
1 1
D. 1+4+16+64+256+ ··· = 1−4 = − ,
3
and so forth. Since each term of series B, except for the first, is greater
than the corresponding term of series A, the sum of series B must be much
more than the sum of series A. Nevertheless, this calculation shows that
series A has an infinite sum, while series B has a negative sum, which is less
than zero, and this is beyond comprehension. Even less can we reconcile