Page 59 - Foundations Of Differential Calculus

42 2. On the Use of Differences in the Theory of Series
These series have a special importance, since they have many applications.
For instance, from these we obtain coefficients of binomials with high de-
grees, and it is clear to anyone with some experience in this area that these
are important.
67. We can also use these to find more easily the partial sums that we
formerly found with differences. Since we found the general term to be of
the form
x − 1 (x − 1) (x − 2) (x − 1) (x − 2) (x − 3)
a + b + c + d + ··· ,
1 1 · 2 1 · 2 · 3
if we take the partial sum of each member and then add all of these, we
will have the partial sum of the series with the given general term. Thus,
since
S.1= x,

1
S. (x − 1) = x (x − 1) ,
2
1
S. (x − 1) (x − 2) = x (x − 1) (x − 2) ,
3
1
S. (x − 1) (x − 2) (x − 3) = x (x − 1) (x − 2) (x − 3) ,
4
and so forth, we have the desired 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
and this is exactly the form we obtained before in paragraph 57 with dif-
ferences.
68. We can also find these partial sums for quotients. Since previously, in
paragraph 34, we obtained, when ω =1,
1 1
Σ = −1 · ,
(x + n)(x + n +1) x + n
so that
1 1 1
S. = −1 · + .
(x + n)(x + n +1) x + n +1 n +1
In a similar way, if we add to the above sum the general term, or, what
comes to the same thing, if in these expressions we substitute x + 1 for x,
then we have
1
S.
(x + n)(x + n +1) (x + n +2)
1 1 1 1
= − · + ·
2 (x + n +1) (x + n +2) 2 (n +1) (n +2)