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.

Since -1 = i^2 I thought that there could be some meaning behind "alternating" series that instead of beginning with (-1)^n begin with (a+b*i)^n, with real coefficients, for abs(a)<1 and abs(b)<1. I'm not sure but it seems that such series are absolutely convergent, because (a+b*i)^n -> 0+0I as n->infinity, hence the term utterly diminishing series instead of alternating series.

As an example,
Where sum((-1)^n*(n^(1/n)-1),n=1..infinity)= 0.187859642462067120248... ,

It still seems that the original post won't accept new replies, so I'm starting a new post.

 It seems I can't add a response to this message, so I added some detail to it.