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In this paper we define a small variation of the Taylor method and a formulafor the global error of this new numerical method that allows us to keep trackof the round-off error and does not require previous knowledge of the exactsolution. As an application we provide a rigorous proof of theconstruction/existence of a periodic solution of the three body problem. Someimages of this periodic motion can be seen athttps://www.youtube.com/watch?v=fSmQyeKcj5k

A small variation of the Taylor Method and periodic solutions of the 3-body problem