Page 53 - Foundations Of Differential Calculus

36 2. On the Use of Differences in the Theory of Series
with X the general term corresponding to the index x, and S the partial
sum. Since S is the sum of the first x terms, the sum of the first x−1 terms
is S −X. Furthermore, X is the difference, since this is what remains when
S − X is subtracted from the next term S.
59. Since X =∆ (S − X) is the difference, as defined in the previous
chapter, provided that we let the constant ω equal 1, if we return to the
sum, then we have ΣX = S − X, and the desired partial sum is

S =ΣX + X + C.

Hence we need to find the sum of the function X by the method previously
discussed, to which we add the general term X to obtain the partial sum.
Since this process involves a constant quantity that must either be added
or subtracted, we need to accommodate this to the present case. It is clear
that if we let x = 0, the number of terms in the sum is zero, and the sum
also should be zero. From this fact the constant C should be calculated by
letting both x = 0 and S = 0. In the expression S =ΣX + X + C with
both S = 0 and x = 0, we obtain the value of C.

60. Since this whole business reduces to the sums of functions we found
(in paragraph 27) when ω = 1, we recall those results, especially for the
powers of x:
0
Σx =Σ1= x,
1 2 1
Σx = x − x,
2 2
2 1 3 1 2 1
Σx = x − x + x,
3 2 6
3 1 4 1 3 1 2
Σx = x − x + x ,
4 2 4
4 1 5 1 4 1 3 1
Σx = x − x + x − x,
5 2 3 30
5 1 6 1 5 5 4 1 2
Σx = x − x + x − x ,
6 2 12 12
6 1 7 1 6 1 5 1 3 1
Σx = x − x + x − x + x.
7 2 2 6 42
In paragraph 29 we gave the sum for a general power of x, provided that
everywhere we let ω = 1. With these formulas we can easily find the partial
sum, provided that the general term is a polynomial in x.
61. Let S.X be the partial sum of the series whose general term is X.As
we have seen,
S.X =ΣX + X + C,