An introduction to the summation of differences of a function; an elementary exposition of the nature of the algebraic processes replaced by the abbreviations of the infinitesimal calculus, by B. F. Groat.

8 SUMMATION OF DIFFERENCES
Whence I -- - - -
I
1 4 4
6 I' 4 2'5 3'6 (n+ I) (I +4)
(n + 2) ( 1+ 5) (n+3) (n +6)
The sum to infinity is 73
1080
x+ 3 5
6. If + _ 5= 3
X ( + I) (x - 2) 4
Suo11 +3 IF n2 + 3 2 + 3 +
SOL.UTIONr+_. (/ +3 1'
1l( -1- I) (? + 2) 2 L"'( 1 + I) (n1 + I) (1- + 2)
2 + = ^-3 --- -+ X x_
12 X(X I)vt 1) + (+ x) (.) + ( )j
- -r2+X_ + 3..x + 2 +3
--.2 +X3 XL- 1+
L2 (X,+ 1) -2 x(+ 1) (n +i )(/-+ 2)
__Ir2 + ( I )(n + 3
2L 2 %(X + I (11 + 1)(12 + 2)2
- - 3 T ) )
2-2 124
'2 2 X+ (/1 + 1)(2 +) 2)
X ~ (1l '}-h ir(1-I- 2 +)
= 2 +
2 L 2 J-l- + I) (n + I) (- + 2) (
5 I ' l + 3
4 2 1 -+ I (2 + I)(1z + 2)
is the sum of n terms. Passing to the limit, we derive the result required.