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.

From the study bench of

Marvin Ray Burns

(317) 371-6571

P.O. Box 19785

Consider the following family of series.

> restart;

> f:=x->sum((-1)^n*(n^(1/n)),n=1..x);

> f1:=x->sum((-1)^n*(n^(1/n)-1),n=1..x);

> f2:=x->sum((-1)^n*(n^(1/n)-2),n=1..x);

The series' have the following sequence of partial sums.

> for i from 3 to 30 do printf("%f, %f, %f\n",evalf(f(i)),evalf(f1(i)),evalf(f2(i))) od;

-1.028036, -0.028036, 0.971964

0.386178, 0.386178, 0.386178
-0.993552, 0.006448, 1.006448
0.354454, 0.354454, 0.354454
-0.966015, 0.033985, 1.033985
0.330824, 0.330824, 0.330824
-0.945694, 0.054306, 1.054306
0.313232, 0.313232, 0.313232
-0.930343, 0.069657, 1.069657
0.299732, 0.299732, 0.299732
-0.918382, 0.081618, 1.081618
0.289060, 0.289060, 0.289060
...

Notice every other partial sum is shared by the three series.

> for i from 1 to 15 do printf("%f, %f, %f\n",evalf(f(2*i)),evalf(f1(2*i)),evalf(f2(2*i))) od;

0.414214, 0.414214, 0.414214
0.386178, 0.386178, 0.386178

0.354454, 0.354454, 0.354454

0.330824, 0.330824, 0.330824
0.313232, 0.313232, 0.313232
0.299732, 0.299732, 0.299732
0.289060, 0.289060, 0.289060
0.280407, 0.280407, 0.280407
0.273242, 0.273242, 0.273242
0.267205, 0.267205, 0.267205
0.262043, 0.262043, 0.262043
0.257574, 0.257574, 0.257574
0.253664, 0.253664, 0.253664
0.250211, 0.250211, 0.250211
0.247138, 0.247138, 0.247138 ...

below, we can see, somewhat, the shared, "every other partial sum," in the following graph, where all of the green, blue gold and red graphs meet.