Page 52 - Foundations Of Differential Calculus

2. On the Use of Differences in the Theory of Series 35
I II III IV V VI
First Differences: b, b , b , b , b , b , b , ...
IV
V
VI
I
II
III
Second Differences: c, c , c , c , c , c , c , ...
II
V
III
IV
I
VI
Third Differences: d, d , d , d , d , d , d , ...
The general term of the series of partial sums, that is, the term correspond-
ing to the index x,is
(x − 1) (x − 2) (x − 1) (x − 2) (x − 3)
0+(x − 1) a + b + c + ··· ,
1 · 2 1 · 2 · 3
but this is also the series of partial sums of the first x−1 terms of the given
I
II
IV
series a, a , a , a ,... .
57. Hence, if instead of x − 1 we write x, we obtain the series of partial
sums
x (x − 1) x (x − 1) (x − 2) x (x − 1) (x − 2) (x − 3)
xa + b + c + d + ··· .
1 · 2 1 · 2 · 3 1 · 2 · 3 · 4
If we let the letters b, c, d, e,... keep the assigned values, then we have
I II III IV V
Series: a, a , a , a , a , a , ...
General Term:
(x − 1) (x − 2) (x − 1) (x − 2) (x − 3)
a +(x − 1) b + c + d
1 · 2 1 · 2 · 3
(x − 1) (x − 2) (x − 3) (x − 4)
+ e + ··· ,
1 · 2 · 3 · 4
Partial Sum:
x (x − 1) x (x − 1) (x − 2) x (x − 1) (x − 2) (x − 3)
xa+ b+ c+ d+··· .
1 · 2 1 · 2 · 3 1 · 2 · 3 · 4
Therefore, once a series of any order is found, the general term can easily
be found from the partial sum in the way we have shown, namely, by
combining differences.
58. This method of finding the partial sum of a series through differences
is most useful for those series whose differences eventually become constant.
In other cases we do not obtain a finite expression. If we pay close attention
to the character of the partial sums, which we have already discussed, then
another method is open to us for finding the partial sum immediately from
the general term. Indeed this method is much more general, and in the
infinite case we obtain finite expressions, rather than the infinite that we
obtain from the previous method. Let a given series be
a, b, c, d, e, f, . . . ,