Page 42 - Foundations Of Differential Calculus
P. 42
2
On the Use of Differences
in the Theory of Series
37. It is well known that the nature of series can be very well illustrated
from first principles through differences. Indeed, arithmetic progressions,
which are ordinarily considered first, have this particular property, that
their first differences are equal to each other. From this it follows that their
second differences and all higher differences will vanish. There are series
whose second differences are constant and for this reason are conveniently
called of the second order, while arithmetic progressions are said to be of
the first order. Furthermore, series of the third order are those whose third
differences are constant; those of the fourth order and higher orders are
those whose fourth and higher differences are constant.
38. In this division there is an infinite number of kinds of series, but by
no means can all series be reduced to one of these. There are innumerably
many series whose successive differences never reduce to constants. Besides
innumerable others, the geometric progressions never have constant differ-
ences of any order. For example, consider
1, 2, 4, 8, 16, 32, 64, 128, ...
1, 2, 4, 8, 16, 32, 64, ...
1, 2, 4, 8, 16, 32, ....
Since the series of differences of each order is equal to the original series,
equality of differences is completely excluded. There are many classes of
series, of which only one class is such that its differences of various orders
