Consider $z_1,...,z_n$ and let $E(z_k) = z_{k+1}$ be a forward shift operator and let $\Delta_-(z_k) = z_k - z_{k-1}$ be a backward difference operator. I'm trying to show that $E = \sum_{j=0}^{\infty}\Delta_-^j$.
I'm just getting started with learning about finite difference operators, and this seems very unintuitive.
I understand so far that $\Delta_- = 1 - E^{-1}$. But this approach makes it seem as though the binomial series above will diverge in some sense. For reference, this is exercise 8.2 in Iserles 'A First Course in the Numerical Analysis of Differential Equations' second edition.