  In this paper we define a small variation of the Taylor method and a formula
for the global error of this new numerical method that allows us to keep track
of the round-off error and does not require previous knowledge of the exact
solution. As an application we provide a rigorous proof of the
construction/existence of a periodic solution of the three body problem. Some
images of this periodic motion can be seen at
https://www.youtube.com/watch?v=fSmQyeKcj5k
