For all real a, the partial sums s(n)= sum((-1)^k (k^(1/k) -a), k=1..n) are bounded so that their limit points form an interval [-1.+ the MRB constant +a, MRB constant] of length 1-a, where the MRB constant is limit(sum((-1)^k*(k^(1/k)), k = 1 ..2*N),N=infinity).
For all complex z, the upper limit point of sn= sum((-1)^k (k^(1/k) -z), k=1..n) is the the MRB constant.
Here is a Demo that compares the first several digits of the MRB constant (MRB) to Pi as far as normality is concerned.
This is an attempt to compute the rationality of MRB. However, even if it computed quadrillions of digits it still would be insufficient for the task. So enjoy playing with it!
The official definition of the MRB constant is found in Mathworld at http://mathworld.wolfram.com/MRBConstant.html.
For a couple of years now, you could read more about it in Wikipedia at
http://en.wikipedia.org/wiki/MRB_constant, and you still can.
However, in the Wikipedia article someone asked for a reference for the irrationality of the MRB constant. At first I thought, I could write an expert or two and they could give me a quick proof of its irrationality; however, when I wrote one of the leading experts on constants about the issue, he told me that it could take a lifetime for someone to prove, and even then the proof might not be right. Thus I figured Ive no time to waste. I started to write down some ideas that might help prove it on mapleprimes at http://www.mapleprimes.com/posts/101425-Could-The-MRB-Constant-Be-Rational.
You might not be too familiar with writing proofs, but a proof is simply an explanation of why some statement is true using only statements that are known to be absolutely true. If on the other hand you think you might be able to help me with it, please do so.
I could use all the help I can get so please consider passing this along, and I thank you in advance.
Marvin Ray Burns
It is likely that almost all mathematics has only been around at most 10,000 years.
At 25 years per generation that comes to only 400 generations of mathematicians, and that's assuming there existed mathematicians in every generation. So humanity has only had at most 400 chances to discover all there is to know about math. So the probability is approximately 1/400 that you potentially could discover something nobody else has. Compared to the lottery those are great odds!
I suggest that you spend a few minutes every day playing with math. Even if you dont discover anything really new, you will be smarter because of it.
To see what happened as I was playing with math, look up MRB constant in almost any search engine.
Marvin Ray Burns
I'm the original investigator of the MRB constant.