Not an answer, just a plot of $F_n(x)$ for $n=2,\ldots,10$:
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[![Fn(x)][1]][1]
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[![Fn(x)2][2]][2]
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  [1]: https://i.sstatic.net/jzWwY.jpg
  [2]: https://i.sstatic.net/c8gFL.jpg