The MRB constant =
Concerning the following divergent and convergent series, we see that
The MRB constant is defined as follows. Consider the sequence of partial sums defined by
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;
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.
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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));
sum((-1)^n*(-1+n^(1/n)), n = 1 .. infinity)) - 0.1878596424620671202485179340542732300559030949;
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