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I will use this post for a list of conclusions drawn from MRB constant N and the many similar approximations that I have found. 

Let x= MRB Constant.   Each approximation is followed by a maple input so you can verify these approximations. 

As some of you know, I'm hoping to, some day, find a closed form expression for the MRB constant.

 Here is my latest little nugget.

Let x=MRB constant.

 = log(x) with an error< ...

The MRB constant is the upper limit point of the sequence of partial sums defined by s(n)= .

Each summand is a real number. However, the function f(n)= is a complex-valued function of a real number, n. This blog is a break in progression of the MRB constant series for the purpose of looking at the "complex" nature of this function. The function can be written in exponential form, .

With this first post I would like to demonstrate, in a Maple document, what happens to f [-2,0). When put together (-1,0) these graphs seem to be describing a hyperbolic spiral. I'm not sure if I'll have more to say, or not. As always, others are welcome to join in.

Download f6142010.mw

If a function is differentiable at some point c of its domain, then it is also continuous at c. However here we extend the notion of differentiability to be valid for individual points on the real number line, specifically positive integers.

 f(n)=(-1)^n* n^(1/n)

THEOREM MRBK 8.0

f=f' / (I*Pi+(1-ln(n))/n^2)| n ∈ {1,2,3,...}

By THEOREM MRBK 4.0, When n is in the set of (positive) integers the derivative of f is exactly I*Pi*f+(1-ln(n))*f/n^2.

So f' = I*Pi*f+(1-ln(n))*f/n^2| n ∈ {1,2,3,...}

Solving for f, we have the following:

f' = I*Pi*f+(1-ln(n))*f/n^2

f' = f*(I*Pi+(1-ln(n))/n^2)

f=f' / (I*Pi+(1-ln(n))/n^2)

For more on this click here (W/A).