The MRB constant = 
Concerning the following divergent and convergent series, we see that
=
and
= 
The MRB constant is defined as follows. Consider the sequence of partial sums defined by
S(n)= sum((-1)^n*n^(1/n),n=1..infinity)
the sequence has two limit points at 0.187859... and 0.187859..-1.
The upper limit point is sometimes known as the MRB constant after my initials.
(See Sloane, N. J. A. Sequences A037077 in "The On-Line Encyclopedia of Integer Sequences.")
Here we will look at the family of divergent and convergent series related to the MRB constant;
in particular
g(x)=(∑) (-1)^n (n^((1)/(n))-x), f(x)=(∑) (-1)^n (n^((x)/(n))-1),
and s(x)=(∑) (-1)^n (n^((x)/(n))-x).
Interstingly, we will see that perhaps g(x)=1/2*x-(1/2-MRB constant) and s(x)-f(x) = 1/2*x-1/2."
First we will graph g(x) and see what closed form best describes it.
g:=x-> 
;
f:=x->
;
s:=x->
;
Download sept042011c.mw for the rest of the document.
Accordingly,
minus plus 
equals .
As shown in Maple by
combine(sum((-1)^n*(n^(1/n)-x), n = 1 .. infinity)-(sum((-1)^n*(n^(x/n)-x), n = 1 .. infinity))+sum((-1)^n*(n^(x/n)-1), n = 1 .. infinity));
simplify((-n^(x/n)+x)*(-1)^n+(-1)^n*(n^(x/n)-1)+(-1)^n*(n^(1/n)-x));
sum((-1)^n*(-1+n^(1/n)), n = 1 .. infinity)) - 0.1878596424620671202485179340542732300559030949;
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