Page 54 - Foundations Of Differential Calculus

2. On the Use of Differences in the Theory of Series 37
where C is the constant we obtain by letting S.X vanish when we let x =0.
It follows that we can express the partial sum of the series of powers, that
n
is, the series for which the general term has the form x .Welet

n
n
n
n
S.X = 1+2 +3 +4 + ··· + x .
Then

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

and so forth.