Thank you Markiyan.

The answer to my question,"As an alalytic extension of the sum is there another value for c such that f(c) = the MRB constant?" is yes. (It is 25.71864....)

Download jan032012.mw

I see; f has a maximium near (22,202)

Download cjan022012.mw

Download bjan022012.mw

Markiyan, It looks like you were showing me where f(c)=c, not necessarily  f(c)=MRB. I wonder if our solution of f(c)=c near c=26 is valid. I mean why would the graph of f have a singularity near 26?

sum(n^(1/n)-1, n = 1 .. infinity) diverges because n^(1/n) - 1 > exp(1/n) - 1 for n >= 3 and

limit((exp(1/n) - 1)/(1/n), n = infinity)=1.

Therefore, sum(exp(1/n)-1, n = 1 .. infinity) diverges and so does sum(n^(1/n)-1, n = 1 .. infinity).

I believe the previous table simply shows some "maxing out" of the software instead of showing convergence of the limit(sum(n^(1/n)-1, n = 1 .. N), N = infinity).

marvinrayburns.com

From the following emperical evidence alone it appears that the limit(sum(n^(1/n)-1, n = 1 .. N), N = infinity)

just might converge.

Here is a table of limits

x    limit(sum(n^(1/n)-1,n=1..N),N=10^x)

1      3.1511647323226333

2     11.452637624960385

3     24.818755742523738

4     43.39893188518323

5     67.26154566874983

6     96.42261224741904

7     130.8850394120835

8     170.649287381816

9     215.7154228449

10   266.08345507

11   321.7533848

12   382.725223

13   448.99902

14   520.57333

15   597.45191

16   679.63721

17   767.0216

18   829.7285

19   829.727

20   829.809

21   829.71

22   829.71

23   829.71

24   829.71

25   829.71

26   829.71

27   829.71

28   829.71

The table is from Mathematica using the following code:

Table[{x,NSum[n^(1/n)-1,{n,1,10^x}, Method

->{"EulerMaclaurin", Method->{"NIntegrate", "MaxRecursion"->20, Method->"DoubleExponential"}}]},{x,28}]

My record high precision computation of the MRB constant is now exactly 300,000 digits to the right of the decimal point.

They were computed from Sat 8 Oct 2011 23:50:40 to Sat 5 Nov 2011 19:53:42.

This run was 0.5766 seconds per digit slower than the 299,998 digit computation even though it used 16GB physical DDR3 RAM on the same machine. The working precision and accuracy goal combination were maximized for exactly 300,000 digits, and the result was automatically saved as a file instead of just being displayed on the front end. Windows reserved a total of 63 GB of working memory of which at 52 GB were recorded being used.

See http://oeis.org/A037077

We did let

g:=x-> ;

f:=x->;

and

s:=x->;

[Comment being edited.]

marvinrayburns.com

We did let s:=x->.

I think it is beautiful how the maximums of the partial sums of s(x) at n= 5, 7, 9, and 11 together with the sum are so separated! I think I will have more to say about the partial sums of s, f and g and how they compare with each other this weekend.

plot([sum((-1)^n*(n^(x/n)-x), n = 1 .. 5), sum((-1)^n*(n^(x/n)-x), n = 1 .. 7), sum((-1)^n*(n^(x/n)-x), n = 1 .. 9), sum((-1)^n*(n^(x/n)-x), n = 1 .. 11), sum((-1)^n*(n^(x/n)-x), n = 1 .. infinity)], x = 1 .. 26, y = -150 .. 250)

My goal here was to identify the best general form to categorize the MRB constant under. Of course it falls under g(x,d)=sum((-1)^n* (n^(x/n)-d)  ,n=1..infinity)  where g(1,1)=MRB constant. However, since I now have the form of Laurent series,  h(z,c,a(n)) =sum((z^n-c)*a(n) ,n=1..infinity), I got thinking that perhaps the MRB constant can best be categorized under the unnamed function f(z,x,d)= sum(z^n*(n^(x/n)-d) ,n=1..infinity); then  f(-1,1,1)=MRB constant.

Any thoughts are appreciated.

Here you can see for yourself how normal the first 5,000 digits of the MRB constant are. However, that doesn't mean that the constant is either normal or irrational. It is fun, however, to see its digits being analyzed so quickly!

Now you can view demonstrations in your browser, after a quick download.

 http://demonstrations.wolfram.com/HowNormalIsTheMRBConstant/

Below we have a table. When we construct large generalized continued fractions from all of the convergents of n repeatedly as in the initial post; it will represent, approximately what is in the table.

{The first value, from 0 by 0.01 to 2, is n and the second is the representation.

{0.01, 100.000000000000000000000000000},

{0.02, 50.0000000000000000000000000000},

{0.03, 1.49999999999999999999983541526},

{0.04, 25.0000000000000000000000000000},

{0.05, 20.0000000000000000000000000000},

{0.06, 8.12126618497817841315615757897},

{0.07, 7.13735628625292704311007524679},

{0.08, 4.22347020820989381899229465580},

{0.09, 1.49999999999999999999395820899},

{0.1, 10.0000000000000000000000000000},

{0.11, 1.49999999999999999999395820899},

{0.12, 1.49999999999999999999395820899},

{0.13, 0.555753104278045912445404118914},

{0.14, 3.27650338501442446313869723500},

{0.15, 3.27650338501442446313869723500},

{0.16, 0.555753104278045912445404118914},

{0.17, 1.49999999999999999999969791045},

{0.18, 0.555753104278045912445404118914},

{0.19, 0.555753104278045912445404118914},

{0.2, 5.00000000000000000000000000000},

{0.21, 0.555753104278045912445404118914},

{0.22, 0.555753104278045912445404118914},

{0.23, 0.555753104278045912445404118914},

{0.24, 0.555753104278045912445404118914},

{0.25, 4.00000000000000000000000000000},

{0.26, 0.555753104278045912445404118914},

{0.27, 0.555753104278045912445404118914},

{0.28, 0.555753104278045912445404118914},

{0.29, 0.555753104278045912445404118914},

{0.3, 1.49999999999999999999999191206},

{0.31, 0.555753104278045912445404118914},

{0.32, 0.555753104278045912445404118914},

{0.33, 0.555753104278045912445404118914},

{0.34, 1.49999999999999999999998489552},

{0.35, 1.49999999999999999999998489552},

{0.36, 0.555753104278045912445404118914},

{0.37, 0.555753104278045912445404118914},

{0.38, 0.555753104278045912445404118914},

{0.39, 0.555753104278045912445404118914},

{0.4, 3.00000000000000000000000000000},

{0.41, 0.555753104278045912445404118914},

{0.42, 0.555753104278045912445404118914},

{0.43, 1.49999999999999999999998489552},

{0.44, 0.555753104278045912445404118914},

{0.45, 1.49999999999999999999997905499},

{0.46, 0.555753104278045912445404118914},

{0.47, 0.555753104278045912445404118914},

{0.48, 0.555753104278045912445404118914},

{0.49, 1.49999999999999999999998489552},

{0.5, 2.00000000000000000000000000000},

{0.51, 0.555753104278045912445404118914},

{0.52, 0.555753104278045912445404118914},

{0.53, 0.555753104278045912445404118914},

{0.54, 0.555753104278045912445404118914},

{0.55, 0.555753104278045912445404118914},

{0.56, 0.555753104278045912445404118914},

{0.57, 0.555753104278045912445404118914},

{0.58, 0.555753104278045912445404118914},

{0.59, 0.555753104278045912445404118914},

{0.6, 0.555753104278045912445404118914},

{0.61, 0.555753104278045912445404118914},

{0.62, 0.555753104278045912445404118914},

{0.63, 0.555753104278045912445404118914},

{0.64, 0.555753104278045912445404118914},

{0.65, 1.49999999999999999999969791045},

{0.66, 0.555753104278045912445404118914},

{0.67, 0.555753104278045912445404118914},

{0.68, 0.555753104278045912445404118914},

{0.69, 0.555753104278045912445404118914},

{0.7, 0.555753104278045912445404118914},

{0.71, 0.555753104278045912445404118914},

{0.72, 0.555753104278045912445404118914},

{0.73, 0.555753104278045912445404118914},

{0.74, 0.555753104278045912445404118914},

{0.75, 1.49999999999999999999395820899},

{0.76, 0.555753104278045912445404118914},

{0.77, 0.555753104278045912445404118914},

{0.78, 0.555753104278045912445404118914},

{0.79, 0.555753104278045912445404118914},

{0.8, 1.49999999999999999999395820899},

{0.81, 0.555753104278045912445404118914},

{0.82, 0.555753104278045912445404118914},

{0.83, 0.555753104278045912445404118914},

{0.84, 0.555753104278045912445404118914},

{0.85, 0.555753104278045912445404118914},

{0.86, 0.555753104278045912445404118914},

{0.87, 0.555753104278045912445404118914},

{0.88, 0.555753104278045912445404118914},

{0.89, 0.555753104278045912445404118914},

{0.9, 0.555753104278045912445404118914},

{0.91, 0.555753104278045912445404118914},

{0.92, 0.555753104278045912445404118914},

{0.93, 0.555753104278045912445404118914},

{0.94, 0.555753104278045912445404118914},

{0.95, 0.555753104278045912445404118914},

{0.96, 0.555753104278045912445404118914},

{0.97, 0.555753104278045912445404118914},

{0.98, 0.555753104278045912445404118914},

{0.99, 0.555753104278045912445404118914},

{1., 1.00000000000000000000000000000},

{1.01, 1.49999999999999999999999894653},

{1.02, 1.49999999999999999999999895275},

{1.03, 1.49999999999999999999999999059},

{1.04, 1.49999999999999999999999894653},

{1.05, 1.49999999999999999999999894653},

{1.06, 1.49999999999999999999999924478},

{1.07, 1.49999999999999999999999924478},

{1.08, 1.49999999999999999999999987039},

{1.09, 1.49999999999999999999999999021},

{1.1, 1.49999999999999999999999900959},

{1.11, 1.49999999999999999999998489552},

{1.12, 1.49999999999999999999999987040},

{1.13, 1.49999999999999999999999924478},

{1.14, 1.49999999999999999999999998956},

{1.15, 1.49999999999999999999999924478},

{1.16, 1.49999999999999999999999998834},

{1.17, 1.49999999999999999999999924478},

{1.18, 1.49999999999999999999999924478},

{1.19, 1.49999999999999999999999924478},

{1.2, 1.49999999999999999999999894653},

{1.21, 1.49999999999999999999999924478},

{1.22, 1.49999999999999999999999924478},

{1.23, 1.49999999999999999999999924478},

{1.24, 1.49999999999999999999999999967},

{1.25, 1.49999999999999999999999999983},

{1.26, 1.49999999999999999999999924478},

{1.27, 1.49999999999999999999999924478},

{1.28, 1.49999999999999999999999924478},

{1.29, 1.49999999999999999999999924478},

{1.3, 1.49999999999999999999999998488},

{1.31, 1.49999999999999999999999924478},

{1.32, 1.49999999999999999999395820899},

{1.33, 1.49999999999999999999999999589},

{1.34, 1.49999999999999999999999924478},

{1.35, 1.49999999999999999999999957716},

{1.36, 1.49999999999999999999999998781},

{1.37, 1.49999999999999999999999999810},

{1.38, 1.49999999999999999999999999800},

{1.39, 1.49999999999999999999999999802},

{1.4, 1.49999999999999999999999997664},

{1.41, 1.49999999999999999999999999892},

{1.42, 1.49999999999999999999999999887},

{1.43, 1.49999999999999999999999924478},

{1.44, 1.49999999999999999999998489552},

{1.45, 1.49999999999999999999999924478},

{1.46, 1.49999999999999999999999999901},

{1.47, 1.49999999999999999999999999930},

{1.48, 1.49999999999999999999999999352},

{1.49, 1.49999999999999999999999924478},

{1.5, 1.49999999999999999999999993553},

{1.51, 1.49999999999999999999999996231},

{1.52, 1.49999999999999999999999996741},

{1.53, 1.49999999999999999999999998359},

{1.54, 1.49999999999999999999999996202},

{1.55, 1.49999999999999999999999996054},

{1.56, 1.49999999999999999999999996054},

{1.57, 1.49999999999999999999999996202},

{1.58, 1.49999999999999999999999996224},

{1.59, 1.49999999999999999999999996224},

{1.6, 1.49999999999999999999998489552},

{1.61, 1.49999999999999999999999996224},

{1.62, 1.49999999999999999999999996224},

{1.63, 1.49999999999999999999999996224},

{1.64, 1.49999999999999999999999996202},

{1.65, 1.49999999999999999999999996054},

{1.66, 1.49999999999999999999999996231},

{1.67, 1.49999999999999999999999997932},

{1.68, 1.49999999999999999999999997841},

{1.69, 1.49999999999999999999999997978},

{1.7, 1.49999999999999999999999998273},

{1.71, 1.49999999999999999999999997976},

{1.72, 1.49999999999999999999999997978},

{1.73, 1.49999999999999999999999997978},

{1.74, 1.49999999999999999999999998024},

{1.75, 1.49999999999999999999999999658},

{1.76, 1.49999999999999999999999996679},

{1.77, 1.49999999999999999999999997986},

{1.78, 1.49999999999999999999999998030},

{1.79, 1.49999999999999999999999997982},

{1.8, 1.49999999999999999999999999999},

{1.81, 1.49999999999999999999999997986},

{1.82, 1.49999999999999999999999997978},

{1.83, 1.49999999999999999999999997987},

{1.84, 1.49999999999999999999999997932},

{1.85, 1.49999999999999999999999997841},

{1.86, 1.49999999999999999999999997932},

{1.87, 1.49999999999999999999999997934},

{1.88, 1.49999999999999999999999997943},

{1.89, 1.49999999999999999999999997943},

{1.9, 1.49999999999999999999999995048},

{1.91, 1.49999999999999999999999997943},

{1.92, 1.49999999999999999999999998241},

{1.93, 1.49999999999999999999999997943},

{1.94, 1.49999999999999999999999997943},

{1.95, 1.49999999999999999999999994659},

{1.96, 1.49999999999999999999999994764},

{1.97, 1.49999999999999999999999997940},

{1.98, 1.49999999999999999999999994751},

{1.99, 1.49999999999999999999999994752},

{2., 2.00000000000000000000000000000}

}

Using the same step size of 0.01, for n from [2,3] we have a representation of 

2.3484074702792301775394210619 for all.

Using the same step size of 0.01, for n from [3,4] we have a representation of 

3.27650338501442446313869723500 for all.

Using the same step size of 0.01, for n from [4,5],[5,6],[7,8],[8,9] we have the same representations shown in infinatenumberof_con.pdf for a step size of 0.1.

Actually there seems to be an infinite number of such constants derived from the repeated use of generalized continued fraction in this manner.

 See the attached PDF:

infinatenumberof_con.pdf .

You don't get that constant from all initial values!

See attached PDF

works_for_some_numbe.pdf