Consider the sequence of divergent series in part evaluated by the following maple input.

f1 := seq((1-a)*(1/2)+sum((-1)^n*(n^(1/n)-a), n = 1 .. infinity), a = 1/10 .. 9*(1/10), 1/10): evalf(f1);

and

f2 := `$`((1-a)*(1/2)+sum((-1)^n*(n^(1/n)-a), n = 1 .. infinity), a = 2 .. 10): evalf(f2);

The Maple output, which is the MRB constant

See the following PDF for the geometry of the MRB constant.

http://www.marvinrayburns.com/what_is_mrb.pdf

If you have any questions, I would like to hear them.

Marvin Ray Burns

In the blog MRB Constant-D I noticed a peculiar outcome to several sets of equations involving f(n) = sin((a+b*floor(n))*Pi/M), where M is a constant to be explored, b is a number to be found and a is a "starting value" that causes f(n) ~=  -1, 0 or 1.

I want to report some progress in finding a closed-form for the MRB constant.

I found a sequence of closed-forms involving the MRB that gives "0." with interesting accuracy far beyond machine precision. To see it for yourself, simply plot 1 + Sin[Pi*(5060936308 + 78389363*Floor[n])/m],
where m is the MRB Constant, for n in any given domain. It is

Download int_vs_sum_for_mr.mw

Looking at the attached worksheet,

it appears that the absolute value, minus 1/2, of the integral of (-1)^x*x^(1/x) from 1 to infinity would equal the partial sum of (-1)^x*x^(1/x) from 1 to where the upper summation is even and growing without bound [0]. Is anyone interested in improving or disproving this conjecture?