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I'm preparing a short presentation on the convergence of the geometric series of matrices, and I'd love some examples of their uses. I've encountered them when approximating inverses of matrices close to $I$, and when computing limiting probabilities of Markov chains. Where else do these series arise?

The audience is a mix of graduate students from pure math, applied math, and statistics. Since mathematical backgrounds are varied, I'll only mention Banach algebras in passing. I won't go into great depth with any examples, but I'd like to justify the topic.

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