Consider the sequence of non-convergent series evaluated by the maple input.

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

and

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

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.

Looking at the attached worksheet,

 Here is an interesting look at the tetrations (infinitely repeating powers) of some fractions.

As a side note, we see that infinitely repeating powers of 1/4 could equal one product of repeating powers of 1/16

. Download 565_multi powers 2.mw

http://www.mapleprimes.com/files/565_multi%20powers%202.mw

This blog is the second extension of MRB Constant.

Some Numerical Qualities Part 1 Please, do not expect me to rehash what I have learned from books, at school or anything from the external links (or refferences to books) that have already been posted in this blog. We intend to write only that which comes from personal investigation, and preferably what is not widely known.

Touching upon numeric methods of computing the MRB Constant

This blog is an extension of MRB Constant.

Some Plot Qualities Part 1 As I post these plot qualities of the MRB Constant. Any input as to triviality will be welcome.

A Spiral Made from the Terms of MRB

>

  To keep the size of this blog from being too large, I broke it into 3 parts: this part, MRB Constant-A and MRB Constant-B. You will see I moved certain parts of this blog to MRB Constant-A and MRB Constant-B. There is also a blog called MRB Constant-C and there may be others as time progresses. I'm sorry that this is so hard to follow; however, it shows some simple Maple usage that may benefit beginners and might inspire others to try harder to discover more.  Finally, for those that are interested in my point of view, it also tells a lot of the history of this constant.
  As of Feb 05, 2010 there also exists blogs titled MRB Constant-D and MRB Constant-E.

In 1998 I felt compelled to research a certain number I felt was over looked. The inspiration actually came from a dream. I quickly began writing friends and telling them I was going to discover a new constant. I first called the constant rc for root constant. Later it became the MRB Constant for the Marvin Ray Burns Constant. My first tools were a Casio programmable graphing calculator and a Sony hand-held "computer-organizer" and a makeshift internet connection. I went to the Inverse Symbolic Calculator Site and used a very old form of Maple. Here is the story surrounding my compulsion: Download 565_final1.doc
From the autobiography in the above final1.doc, you see that I experienced something mystical while researching that constant with maple. As you read in final1.doc, I am a tradesman; however, the intoxicating power of numbers left me hung-over for knowledge. Particularly, there was the Irresistible draw on my mind that there could be something special to be discovered about that constant I mentioned in the last paragraph. It was in my search for something special about that constant,from an alternating series, that I enrolled in college as a 40 year old.
However, in my second semester of calculus, I got the sad impression from my books that that many alternating series do not converge and thus you can not rely on them to give you any particular values. These series do not converge. They have nothing to do with convergence. They are valueless (have no defined sum). Having already explored some of those alternating series, I was left with a bitter taste in my mouth. Here is a worksheet about a family of so called "non converging" alternating series’ that we can rely on to converge upon at least one constant value. Download 565_mrbgraphs.mws

For more information of the constant shown in the above worksheet you may study this file.
It has a link to a third part that might be of some interest.

Download 565_QA-1.doc
I make no claim to rigor or expertise; the above is just my findings thus far. Click the above MRB Constant Link to continue reading marvinrayburns.com

A couple of years ago I did the following work on Trigonometric integration.
Does anyone have any improvements to add to it? I would like to see it expanded to all complex numbers. I'll soon be working on it again.

Download 565_Pre_exp[1].doc
View file details

My Desire