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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.

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:

The diagonalization in part (b) gives $M^k=P\mathrm{diag}(a^k,b^k,c^k)P^{-1}$. HenceSufficiency follows from the scalar powers tending to zero; necessity also follows immediately from the diagonal entries of $M^k$. Under these conditions the matrix geometric series converges, and its sum isWriting $A_0=1/(1-a)$, $B_0=1/(1-b)$ and $C_0=1/(1-c)$, all entries are explicitlyFor example, $B_0-A_0=(b-a)/[(1-a)(1-b)]$ and $C_0-B_0=(c-b)/[(1-b)(1-c)]$. These expressions remain valid when parameter values coincide.

Generate successive matrix powers and take cumulative sums using matrix addition:For the stipulated square matrix, iterate (multMat m) m produces m, m*m, m*(m*m), and so on. Associativity of exact matrix multiplication identifies these as $M,M^2,M^3,\dots$. The left scan scanl1 addMat then returnsAll dimensions remain compatible. The infinite outer list is available a finite prefix at a time under lazy evaluation; no convergence of the matrix geometric series is needed. With floating-point entries, the expression specifies this particular multiplication and addition order, since machine arithmetic need not obey the exact algebraic laws.

If $B^k=0$ with $k\ge 1$, put $S=I+B+\cdots +B^{k-1}$. The matrix geometric series telescopes on both sides:Thus $I-B$ is nonsingular, andThis is a finite polynomial identity for a nilpotent matrix; no convergence argument or bound $\Vert B\Vert <1$ is required.