I'm reading a book (Digital Signal Processing by Jonathan Steyn) which defines the infinite sum:
$$r\equiv\sum^\infty_{m=0}z^{-m}x_n$$
This is just the sum of $x_1 + x_2 + x_3 +\dots +x_n$ to infinty. Then it defines the finite difference as $(z^{-1}$ being the previous value):
$$\Delta\equiv(1-z^{-1})$$
Then it claims that they are related such that the one applied to the other equals the identity operator. By referring to the telescoping series $x_1 - x_0 + x_2 - x_1\dots x_n - x_{n-1} = x_n - x_0$, they say you should be able to prove this.
However, this has me stumped. If you take the series $xk = 1,2,3,\dots$ Then you get $x_n - x_0 = x_n - 1$. How can this be the identity operator?