Page 29 - Foundations Of Differential Calculus

12 1. On Finite Differences
In a similar way we let
2 3
−1 ω ω ω
∆.x = − + − + ··· ,
x 2 x 3 x 4
−2 2ω 3ω 2 4ω 3
∆.x = − + − + ··· ,
x 3 x 4 x 5

−3 3ω 6ω 2 10ω 3
∆.x = − + − + ··· ,
x 4 x 5 x 6
2 3
−4 4ω 10ω 20ω
∆.x = − + − + ··· .
x 5 x 6 x 7
We continue in the same way for the rest. For fractions we have

1/2 ω ω 2 ω 3
∆.x = − + − ··· ,
2x 1/2 8x 3/2 16x 5/2
1/3 ω ω 2 5ω 3
∆.x = − + − ··· ,
3x 2/3 9x 5/9 81x 8/3
−1/2 ω 3ω 2 5ω 3
∆.x = − + − + ··· ,
2x 3/2 8x 5/2 16x 7/2
2 14ω 3
2ω
ω
∆.x −1/3 = − + − + ··· .
3x 4/3 9x 7/3 81x 10/3
18. It should be clear that if the exponent is not a positive integer, then
these differences will progress without limit, that is, there will be an infinite
number of terms. Nevertheless, these same differences can be expressed by
I
a finite expression. If we let y = x −1 =1/x, then y =1/ (x + ω), so that
−1 1 1 1
∆.x =∆. = − .
x x + ω x
Hence, if the fraction 1/ (x + ω) is expressed as a series, then we obtain the
infinite expression we saw before. In a similar way we have

−2 1 1 1
∆.x =∆. = 2 − .
x 2 (x + ω) x 2
Furthermore, for irrational expressions we have
√ √ √
∆. x = x + ω − x
and
1 1 1
∆.√ = √ − √ .
x x + ω x