Page 58 - Foundations Of Differential Calculus

2. On the Use of Differences in the Theory of Series 41
section and the sections following. When we let ω = 1,wehave

1
Σ(x + n)= (x + n − 1) (x + n) ,
2
1
Σ(x + n)(x + n +1) = (x + n − 1) (x + n)(x + n +1) ,
3
1
Σ(x + n)(x + n +1) (x + n +2) = (x + n − 1) (x + n)(x + n +1)
4
× (x + n +2) ,

and so forth. If we add to these sums the general term and a constant,
which can be calculated by letting x = 0, with the partial sum vanishing,
then we obtain the following:
1 1
S. (x + n)= (x + n)(x + n +1) − n (n +1) ,
2 2
1
S. (x + n)(x + n +1) = (x + n)(x + n +1) (x + n +2)
3
1
− n (n +1) (n +2) ,
3
1
S. (x + n)(x + n +1) (x + n +2) = (x + n)(x + n +1) (x + n +2)
4
× (x + n +3)
1
− n (n +1) (n +2) (n +3) ,
4

and so forth.
If we let n =0 or n = −1, then the constant in these partial sums will
vanish.
66. For the series 1, 2, 3, 4, 5,... , whose general term is x, the partial sum
1
is x (x +1) . The series of partial sums, 1, 3, 6, 10, 15,... has a partial sum
2
x (x +1) (x +2)
.
1 · 2 · 3
This series of partial sums 1, 4, 10, 20, 35,... has a partial sum
x (x +1) (x +2) (x +3)
,
1 · 2 · 3 · 4
which is the general term of the series 1, 5, 15, 35, 70,... with partial sum
x (x +1) (x +2) (x +3) (x +4)
.
1 · 2 · 3 · 4 · 5