Page 51 - Foundations Of Differential Calculus

34 2. On the Use of Differences in the Theory of Series
first term of the series of partial sums, to give

I II III IV V
0, A, A , A , A , A , A , ...,
then the series of first differences is the given series
IV
V
III
I
II
a, a , a , a , a , a , ....
55. For this reason, the first differences of the given series are the second
differences of the series of partial sums; the second differences of the former
are the third differences of the latter; the third of the former are the fourth
of the latter, and so forth. Hence, if the given series finally has constant
differences, then the series of partial sums also eventually has constant
differences and so is of the same kind except one order higher. It follows
that this kind of series always has a partial sum that can be given as a
finite expression. Indeed, the general term of the series
I II III IV
0, A, A , A , A , A , ...,
or that expression which corresponds to x, gives the sum of the x−1 terms
I
II
IV
III
of the series a, a , a , a , a ,... . If instead of x we write x+1, we obtain
the sum of x terms which is the general term.
56. Let a given series be
I II III IV V VI
a, a , a , a , a , a , a , ....
The series of first differences is

I II III IV V VI
b, b , b , b , b , b , b , ...,
the series of second differences is
I II III IV V VI
c, c , c , c , c , c , c , ...,
the series of third differences is
I II III IV V VI
d, d , d , d , d , d , d , ...,

and so forth, until we come to constant differences. We then form the series
of partial sums, with 0 as its first term, and the succeeding differences in
the following way:
Indices: 1, 2, 3, 4, 5, 6, 7, ...
III
V
IV
I
II
Partial Sums: 0, A, A , A , A , A , A , ...
I II III IV V VI
Given Series: a, a , a , a , a , a , a , ...