Matrix geometric series ($(I-D){\sum }_{j=0}^m{D}^j=I-D^{m+1}$)

The finite matrix geometric series telescopes as stated in the title. If its partial sums converge, then their successive differences show $D^{m+1}\to 0$. Passing to the limit gives $(I-D){\sum }_{j\ge 0}{D}^j=I$, 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.