Page 41 - Foundations Of Differential Calculus

24 1. On Finite Differences
Example 1. Find the function whose difference is
3x +2ω
.
x (x + ω)(x +2ω)

The given difference is expressed as partial fractions:
1 1 1 1 2 1
· + · − · .
ω x ω x + ω ω x +2ω
From the previous formula
1 1 1
Σ =Σ − ,
x + nω x +(n +1) ω x + nω
we have
1 1 1
Σ =Σ − .
x x + ω x
It follows that the desired sum is
1 1 1 1 2 1 2 1 2 1 1
Σ + Σ − Σ = Σ − Σ − .
ω x ω x + ω ω x +2ω ω x + ω ω x +2ω ωx
But
1 1 1
Σ =Σ − ,
x + ω x +2ω x + ω
so that the desired sum is
1 2 −3x − ω
− − = .
ωx ω (x + ω) ωx (x + ω)
Example 2. Find the function whose difference is
3ω
.
x (x +3ω)

We let this difference be z. Then
1 1
z = −
x x +3ω
and
1 1 1 1 1
Σz =Σ − Σ =Σ − Σ −
x x +3ω x + ω x +3ω x
1 1 1 1 1 1 1
=Σ − Σ − − = − − − ,
x +2ω x +3ω x x + ω x x + ω x +2ω
which is the desired sum. Whenever the signs of the sums finally cancel each
other, we will be able to find the sum. However, if this mutual annihilation
does not occur, it signifies that this sum cannot be found.