Geometric Series of Discounted Matrices

Question

Does the following series have a closed form solution for scalar $\gamma$ and vector $x$ such that $0 \leq \gamma < 1$?

$$ (M + \gamma M + \gamma^2M^2 + \gamma^3M^3 +\ldots + \gamma^\infty M^\infty )x $$

Much like how the scalar version does?

Answer 1

If $\|M\|\leq 1$, then we have $(I + \gamma M + \gamma^2M^2 + \gamma^3M^3 +\ldots)=(I-\gamma M)^{-1}$. So a closed form solution would be $$\left((M-I)+(I-\gamma M)^{-1}\right)x=Mx-Ix+(I-\gamma M)^{-1}x$$ (given that $\|\gamma M\|<1$).