and you're looking for the value of
One can show via reverse induction over that . So in the limit, defining , we have
Using interval arithmetic we can then obtain rigorous bounds on . Define the interval-valued function
where the usual interval arithmetic rules apply,
(because all our intervals are positive, except which never appears in a denominator). It should be possible to show that for all , and so . Assuming that's true,
narrows down the desired number to significant digits.
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