Page 74 - Foundations Of Differential Calculus

3. On the Infinite and the Infinitely Small 57
examples of phenomena of this kind are found in the theory of plane curves,
1
treated in a preceding book.
100. In the same way infinite terms often occur in series. Thus, in the
harmonic series, whose general term is 1/x, the term corresponding to the
index x = 0 is the infinite term 1/0. The whole series is as follows:

1 1 1 1 1 1 1 1
..., − , − , − , − , + , + , + , + , ....
4 3 2 1 0 1 2 3
Going from right to left the terms increase, so that 1/0 is infinitely large.
Once it has passed through, the terms become decreasing and negative.
Hence, an infinitely large quantity can be thought of as some kind of limit,
passing through which positive numbers become negative and vice versa.
For this reason it has seemed to many that the negative numbers can be
thought of as greater than infinity, since in this series the terms continu-
ously increase, and once they have reached infinity, they become negative.
2
However, if we consider the series whose general term is 1/x , then after
passing through infinity, the terms become positive again,
1 1 1 1 1 1 1
..., + , + , + , + , + , + , + , ...,
9 4 1 0 1 4 9
and no one would say that these are greater than infinity.
101. Frequently, in a series an infinite term will constitute a limit sep-
arating real terms from complex, as occurs in the following series, whose
√
general term is 1/ x:
1 1 1 1 1 1 1
..., +√ , +√ , +√ , + , +√ , +√ , +√ , ....
−3 −2 −1 0 1 2 3
From this it does not follow that complex numbers are greater than infinity,
since from the series previously discussed,
√ √ √ √ √ √
..., + −3, + −2, + −1, +0, + 1, + 2, + 3, ...,
it would equally follow that the complex numbers are less than zero. It is
possible to show a change from real terms to complex, where the limit is
√
neither 0 nor ∞, for example if the general term is 1 + x. In these cases,
due to the irrationality, each term has two values. In the limit between
real and complex numbers the two values always come together as equals.
Nevertheless, whenever there are terms that are first positive and then
become negative, the transition is always through a limit that is infinitely
small or infinitely large. This is all due to the law of continuity, which is
most clearly seen through plane curves.

1 Introduction,BookII.