Viète à grande Vitesse or Accelerated Viète (kind of)

[EdS2]

Thanks, Gerson and Albert, for your explanations.

I'm trying to understand in what way these processes are justified - they are evidently very effective. Is there a leap of faith or an element of guessing going on? Is it perhaps the case that there's infinite variety of terms (or factors) which would be valid - so long as a term tends to zero or a factor tends to one, perhaps, as in both cases that makes it valid. So we're in a search space where we're mainly seeking simplicity. Simpler is better. But also, some choices will make for a better acceleration - what's the measure of that?

Why is it valid to look at numbers which seem to be converging on an integer, and then choose that integer? Perhaps this is obvious!

I ask not in a spirit of attacking, of course, but a spirit of trying to understand. This seems to be a branch of experimental mathematics.