= Matrix geometric series
{title2=$(I-D)\sum_{j=0}^mD^j=I-D^{m+1}$}
The finite <matrix geometric series> telescopes as stated in the title. If its partial sums converge, then their successive differences show $D^{m+1}\to0$. Passing to the limit gives $(I-D)\sum_{j\geq0}D^j=I$, so the sum is the <matrix inverse> $(I-D)^{-1}$. Conversely, existence of the inverse alone does not imply series convergence: the scalar matrix $D=2I$ is a counterexample. The <Neumann series> gives a sufficient condition from a matrix norm smaller than one.
For a <nilpotent matrix> with $D^r=0$, convergence is automatic and the inverse is the finite polynomial $I+D+\cdots+D^{r-1}$. Multiplying by $I-D$ directly verifies this inverse without computing a determinant or diagonalizing.
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