Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2006 / Paper 1 / 1 / c / Solution

Let $S_m=I+D+\cdots +D^m$. The finite matrix geometric series telescopes to $(I-D)S_m=I-D^{m+1}$. By the assumed convergence, its terms $D^{m+1}$ tend to zero, so the limit $S$ satisfies $(I-D)S=I$. Consequently $S=(I-D{)}^{-1}$. This is the finite-dimensional Neumann series argument; invertibility by itself would not imply convergence.

For the stated nilpotent matrix, $D^3=0$ and only the first three terms remain: