Page 37 - Foundations Of Differential Calculus

20 1. On Finite Differences
29. If we carefully observe the sums of the powers of x that we have found,
the first, second, and third terms, we quickly discover the laws of formation
that they follow. The law for the following terms is not so obvious that we
n
can state in general the sum for the power x . Later (in paragraph 132 of
the second part) we will show that

x n+1 1 n 1 nω n−1 1 n (n − 1) (n − 2) ω 3 n−3
n
Σx = − x + · x − · x
(n +1) ω 2 2 2 · 3 6 2 · 3 · 4 · 5
5
1 n (n − 1) (n − 2) (n − 3) (n − 4) ω n−5
+ · x
6 2 · 3 · 4 · 5 · 6 · 7
3 n (n − 1) ··· (n − 6) ω 7 n−7
− · x
10 2 · 3 ··· 8 · 9
9
5 n (n − 1) ··· (n − 8) ω n−9
+ · x
6 2 · 3 ··· 10 · 11
691 n (n − 1) ··· (n − 10) ω 11 n−11
− · x
210 2 · 3 ··· 12 · 13
35 n (n − 1) ··· (n − 12) ω 13 n−13
+ · x
2 2 · 3 ··· 14 · 15
15
3617 n (n − 1) ··· (n − 14) ω n−15
− · x
30 2 · 3 ··· 16 · 17
43867 n (n − 1) ··· (n − 16) ω 17 n−17
+ · x
42 2 · 3 ··· 18 · 19
1222277 n (n − 1) ··· (n − 18) ω 19 n−19
− · x
110 2 · 3 ··· 20 · 21
21
854513 n (n − 1) ··· (n − 20) ω n−21
+ · x
6 2 · 3 ··· 22 · 23
1181820455 n (n − 1) ··· (n − 22) ω 23 n−23
− · x
546 2 · 3 ··· 24 · 25
76977927 n (n − 1) ··· (n − 24) ω 25 n−25
+ · x
2 2 · 3 ··· 26 · 27
27
23749461029 n (n − 1) ··· (n − 26) ω n−27
− · x
30 2 · 3 ··· 28 · 29
8615841276005 n (n − 1) ··· (n − 28) ω 29 n−29
+ · x
462 2 · 3 ··· 30 · 31
+ ··· + C.

The main interest here is the sequence of purely numerical coefficients. It
is not yet time to explain how these are formed.