Here is the progress made in the investigation of what I call the convergents constants:
https://oeis.org/wiki/Table_of_convergents_constants
I wonder if anyone would be interested in adding anything to it. I would like to see the convergents constants studied some in Maple to compare with my Mathematica results; my investigation is in dire need of some proof other than my empirical evidence and my study has at least one major open question!
