Page 48 - Foundations Of Differential Calculus

2. On the Use of Differences in the Theory of Series 31
and so forth, provided that n is an integer and is larger than the order of the
given series. These series can also arise from fractions whose denominator is
n 1
(1 − y) , as is shown in a previous book where recurrent series are treated
at greater length.
48. As we have seen, every series of this class, no matter of what order,
has a general term that is a polynomial. On the other hand, we will see that
every series that has this kind of function for its general term belongs to
this class of series and the differences eventually are a constant. Indeed, if
2
the general term is a second-degree polynomial of the form ax +bx+c, then
the series obtained by letting x be equal successively to 1, 2, 3, 4, 5,... , will
be of the second order, and the second difference will be constant. Likewise,
3
2
if the general term is a third-degree polynomial ax + bx + cx + d, then
the series will be of the third order, and so forth.
49. From the general term we can find not only all of the terms of the
series, but also the series of differences, both the first differences and also
the higher differences. If the first term of a series is subtracted from the
second, we obtain the first term of the series of differences. Likewise, we
obtain the second term of this series if we subtract the second term from
the third term of the original series. Thus we obtain the term of the series
of differences whose index is x if we subtract the term of the original series
whose index is x from the term whose index is x + 1. Hence, if in the
general term of the series we substitute x + 1 for x and from this subtract
the general term, the remainder will be the general term of the series of
differences. If X is the general term of the series, then its difference ∆X
(which is obtained in the way shown in the previous chapter if we let ω =1)
2
is the general term of the series of first differences. Likewise, ∆ X is the
3
general term of the series of second differences; ∆ X is the general term of
the series of third differences, and so forth.

50. If the general term X is a polynomial of degree n, from the previous
chapter we know that its difference ∆X is a polynomial of degree n − 1.
2 3
Furthermore, ∆ X is of degree n − 2, and ∆ X is of degree n − 3, and so
forth. It follows that if X is a first-degree polynomial, as is ax + b, then its
difference ∆X is the constant a. Since this is the general term of the series
of first differences, we see that a series whose general term is a first-degree
polynomial is an arithmetic progression, that is, a series of the first order.
2
Likewise, if the general term is a second-degree polynomial, since ∆ X is a
constant, the series of second differences is constant and the original series
is of second order. In like manner it is always true that the degree of the
general term is the order of the series it defines.

1 Introduction, Book I, Chapter IV.