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Is the Neumann series of a real semidefinite matrix asymptotic?

Let $A \in \mathbb{R}^{m \times m}$ be a real symmetric negative-semidefinite matrix and consider the Neumann series $$\sum_{k=0}^\infty t^kA^k = I + tA + t^2 A^2 + \cdots$$ where $t > 0$ is a ...