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

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

Since 1998, the basic numerical example of summing the actual terms of the MRB Constant has been as follows:

This Iterated Exponential Constant starts out simple.
Consider one.
Now if I was to ask you what the first root of one is,
let's just say the answer is 1.
The second step is to consider the square root of two.
That is an irrational number a little greater than 1.4142.
Third, consider the cube root of three.
Now we also want the fourth root of four, fifth root of five, and so on.
Now after we get all of these we want to add them in a an alternating series, as follows
1^(1/1)-2^(1/2)+3^(1/3)- 4^(1/4)+5^(1/5)
What do you get on your calculator as you work the series as far as I showed above?
You should get a positive number slightly greater than .99.
1^(1/1)-2^(1/2)+3^(1/3)- 4^(1/4)+5^(1/5)-6^(1/6)
Now, what do you get on your calculator as you work the series as far as I showed above?
You should get a negative number slightly less than -.354454.
We know what is happening here; don't we?
The series is "alternating" as its sum branches off from the "first root of one" into two separate convergent sums.
We have two series. One has an even last term, the other odd.

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

Beginning in this post, we will consider the graph of the terms of f1, which terms we treat as a continuous function g1.

g1 Makes the terms of f1(ie. the absolute convergent form of the MRB Constant)
Within special ranges, the sequence, g1, has an interesting spiral design that finds itself a center at the origin.

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