Page 76 - Foundations Of Differential Calculus

3. On the Infinite and the Infinitely Small 59
with ordinary ideas the results of this and the following series C, D, and
so forth, which have negative sums while all of the terms are positive.
104. For this reason, the opinion suggested above, namely, that negative
numbers might sometimes be considered greater than the infinite, that is,
more than infinity, might seem to be more probable. Since it is also true
that when decreasing numbers go beyond zero they become negative, a
distinction has to be made between negative numbers like −1, −2, −3,...
and negative numbers like
+1 +2 +3
, ...,
−1 −1 −1

the former being less than zero and the latter being greater than the infinite.
Even with this agreement, the difficulty is not eliminated, as is suggested
by the following series:
2 3 4 1
1+2x +3x +4x +5x + ··· = 2 ,
(1 − x)

from which we obtain the following series:
1 1
A. 1+2+3+4+5+ ··· = 2 = = ∞,
(1−1) 0
1
B. 1+4+12+32+80+ ··· = 2 =1.
(1−2)
Now, every term of series B is greater than the corresponding term of series
A, except for the first term, and insofar as the sum of series A is infinite,
while the sum of series B is equal to 1, which is only the first term, the
suggested principle is no explanation at all.

105. Since if we were to deny that
+1 +a −a
−1= and = ,
−1 −b +b
the very firmest foundations of analysis would collapse, the previously sug-
gested explanation is not to be admitted. We ought rather to deny that the
sums that the general formula supplied are the true sums. Since these series
are derived by continual division, and while the remainders are divided fur-
ther, the remainders always grow larger the longer we continue, so that the
remainder can never be neglected. Even less can the last remainder, that is,
that divided by an infinitesimal, be omitted, since it is infinite. Since we did
not observe this in the previous series where the remainder became zero, it
should not be surprising that those sums led to absurd results. Since this
response is derived from the very origin of the series itself, it is most true
and it removes all doubt.