Page 46 - Foundations Of Differential Calculus

2. On the Use of Differences in the Theory of Series 29
I II III IV V
Terms: a, a , a , a , a , a , ...

III
IV
I
II
First Differences: b, b , b , b , b , ...
I II III
Second Differences: c, c , c , c , ...
I
II
Third Differences: d, d , d , ...
I
Fourth Differences: e, e , ...

Fifth Differences: f, ...

and so forth. From the first term of the series and the first terms of the
differences, b, c, d, e, f,... we can express the general term as

(x − 1) (x − 1) (x − 2) (x − 1) (x − 2) (x − 3)
a + b + c + d
1 1 · 2 1 · 2 · 3
(x − 1) (x − 2) (x − 3) (x − 4)
+ e + ···
1 · 2 · 3 · 4

until we come to the constant differences. From this it is clear that if we
never produce constant differences, then the general term will be expressed
by an infinite series.
45. Since the differences are formed from the terms of the given series, if
these values are substituted, the general term in this form for any series
of the first, second, and third orders have been given. For a series of the
fourth order the general term is

(x − 1) (x − 2) (x − 3) (x − 4) (x − 5)
1 · 2 · 3 · 4
a 4a 6a 4a a
IV III II I
× − + − + .
x − 5 x − 4 x − 3 x − 2 x − 1
From this the law of formation for the general term for higher-order se-
quences is easily seen. It is also clear that for any order the general term
will be a polynomial in x whose degree will be no higher than the or-
der of the series to which it refers. Thus, a series of the first order has
a general term that is a first-degree function, a second-order series has a
second-degree term, and so forth.