= 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:
$$
\boxed{(I-D)^{-1}=I+D+D^2=\begin{pmatrix}1&1&1\\0&1&1\\0&0&1\end{pmatrix}.}
$$
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