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... ,
sum((1/3-1/2*I)^n*(n^(1/n)-1),n=1..infinity) = -.164536552011234424905245789861...-0.0928948452694530168658278207290...*I

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