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As defined at http://mathworld.wolfram.com/MRBConstant.html the MRB constant is

The MRB constant =    

Concerning the following divergent and convergent series, we see that

=

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...

I've made a little progress in definig a new set of constants from the generalized continued fraction that I mentioned in my previos post.

See https://oeis.org/wiki/Convergents_constant.