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For questions related to the various policy evaluation (PE) algorithms, which are numerical iterative algorithms that are used to find the value function associated with a given policy, which is often denoted as the "prediction problem". PE is also considered a dynamic programming method, which is regularly discussed in reinforcement learning textbooks.

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Is the existence and uniqueness of the state-value function for $\gamma < 1$ theoretical?

In essence, your question is about convergence of infinite series. The mathematical discipline that studies such series is hundreds (if not thousands) years old an has nothing to do with "hardware arc …
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