Page 44 - Foundations Of Differential Calculus

2. On the Use of Differences in the Theory of Series 27
both terms of the series itself and terms of the series of differences. If we
I I
call the second term of the series a , since b = a −a, then the general term
is
I I I
a − a x +2a − a = a (x − 1) − a (x − 2) .
Hence from our knowledge of the first and second terms of an arithmetic
progression we form the general term.

42. Let a be the first term of a series of the second order, let b be the first
term of the series of first differences, and let c be the first term of the series
of second differences. Then the series with its differences have the following
form:

Indices:
1, 2, 3, 4, 5, 6, 7
Terms:

a, a + b, a +2b + c, a +3b +3c, a +4b +6c, a +5b +10c, a +6b +15c

First Differences:
b, b + c, b +2c, b +3c, b +4c, b +5c, ...

Second Differences:
c, c, c, c, c, . . .

By inspection we conclude that the term with index x will be

(x − 1) (x − 2)
a +(x − 1) b + c,
1 · 2
and this is the general term of the given series. However, if we let the
I
I
II
second term of the series be a and the third term be a , since b = a − a
I
II
and c = a − 2a + a, as we understand from the definition of differences
(paragraph 10), we have the general term
I (x − 1) (x − 2) II I
a +(x − 1) a − a + a − 2a + a .
1 · 2
But this reduces to the form
II
I
a (x − 1) (x − 2) 2a (x − 1) (x − 3) a (x − 2) (x − 3)
− + ,
1 · 2 1 · 2 1 · 2
or
a II 2a I a
(x − 1) (x − 2) − (x − 1) (x − 3) + (x − 2) (x − 3) ,
2 2 2