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We present a method for the solution of the Dynamic Programming Equations (DPE), the discrete-time HJB PDE that arises in an infinite time optimal control problem. The method exploits the idea of Al'brecht in the continous-time systems in locally approximating the solution. We explicitly find the power series expansion of the optimal control and cost in a term by term fashion. We also prove the local existence of these solutions by using the discrete-time version of the Pontryagin Maximum Principle and the local stable manifold theorem.