Advanced Search
All Posts : MaplePrimes Commons General Technical Discussions

The MRB Constant-B

Marvin Ray Burns 550
February 05 2007
0

This blog is the second extension of MRB Constant. On May 8, 2012 I replaced some missing parts of this blog.

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. I 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.

 

Next we view how slow the series' are to converge.

> 

g:=x->(-1)^x*(x^(1/x)):
for a from 1 to 5 do printf("10^ %a terms gives a difference of %a\n",a,evalf(m-sum(g(x),x=1..10^a))) od;

10^ 1 terms gives a difference of -.125372118
10^ 2 terms gives a difference of -.23469899e-1

10^ 3 terms gives a difference of -.3464337e-2

10^ 4 terms gives a difference of -.460656e-3

10^ 5 terms gives a difference of -.57971e-4

Further evaluation reveals:

 

Let m stant for the mrb constant.
    
m=0.1878596424620671202485179340542732300559030949




The partial sum up to 10 terms:
 0.3132317592546559751298942294
giving a difference from m of
-0.125372116792588854881376295

The partial sum up to 10^2 terms:
0.2113295433469410694850358682
giving a difference from m of
-0.0234699008848739492365179341


The partial sum up to 10^3 terms:
0.2113295433469410694850358682
giving a difference from m of
-0.0234699008848739492365179341


The partial sum up to 10^4 terms:
0.1883203510769505046388977903
giving a difference from m of
-0.000460708614883384390379856


The partial sum up to 10^5 terms:
0.1879172101403520645260206346
giving a difference from m of
-0.000057567678284944277502701


The partial sum up to 10^6 terms:
0.1878665502618594832229218275
giving a difference from m of
-6.907799792362974403893*10^-6


The partial sum up to 10^7 terms:
0.1878604483674613557319422890
giving a difference from m of
-8.0590539423548342435*10^-7


The partial sum up to 10^8 terms:
0.1878597345654788875307441677
giving a difference from m of
-9.210341176728222623*10^-8


The partial sum up to 10^9 terms:
0.1878596528237001411543445189;m-ff
giving a difference from m of
-1.036163302090582658*10^-8


The partial sum up to 10^10 terms:
0.1878596436133596680159507666;m-ff
giving a difference from m of
-1.15129254776743283*10^-9

 

> 

 


Finally, we will re-define a series and look at a few shortcuts to approximating the value of the MRB Constant:

> 

g1:=x->(-1)^x*(x^(1/x)-1):

m := evalf(sum(g1(x), x = 1 .. infinity))

m := .1878596425

The area under the curve from 1-0.22 to 1 gives a better approximation to m than 10^4 terms:

> 

 

m-evalf(abs(int(g(x), x = .78 .. 1)))

-0.3824710e-3

The area under the curve from 1 to 2+0.22 gives a better approximation to m than 10^5 terms:

m-evalf(1-abs(int(g(x), x = 1 .. 2.2)))

0.380424e-4

The area under the curve from 2+0.22 to 2+0.331887 gives a better approximation to m than 10^7 terms:

m-evalf(abs(int(g(x), x = 2.2 .. 2.331887)))

0.5478e-6

> 

 

 

 

 

 

 

Download 565_Dec_26.mw

 

-

Some Numerical Qualities Part 2 I am not sure if the following is trivial; nonetheless, here is my findings for the day: As an afterthought, what I say below, in this message, about q is also true about g1 and nearly its whole familly. This includes #1=x->(-1)^x*(x^(1/x)-I) i.e. function familly members in the complex. The following also applies to distant relatives like, #2=x->(-1)^x*(x^(a/x)-c), where a IS A REAL NUMBER and c is any complex constnat.

 

Earlier we defined g1 and m.

> g1:=x->(-1)^x*(x^(1/x)-1):

> m:=evalf(sum(g1(x),x=1..infinity));

 

 

Now we will define the function q -- as a member of the sequences that produce the series' known for producing the MRB Constant (see the initial entry for this blog) -- but also has the constant in itself.

> q:=x->(-1)^x*(x^(1/x)-m):

 

 

Earlier we defined g1 and m.

> 

g1:=x->(-1)^x*(x^(1/x)-1):
> 

m:=evalf(sum(g1(x),x=1..infinity));

m := .1878596425

Now we will define the function q -- as a member of the sequences that produce the series' known for producing the MRB Constant (see the initial entry for this blog) -- but also has the constant in itself.

> 

q:=x->(-1)^x*(x^(1/x)-m):

Im(q(x))= Re(q(x)) at 1/4 +1*n | n is an integer!

> 

plot(abs(q(x)),x=0..2);plot([Re(q(x)),Im(q(x))],x=0..2);

> 

for a from 1 to 7 do evalf(eval(Im(q(x))-Re(q(x)),x=a+1/4)) od;
> 

 

0.

0.

0.

0.

0.

0.

0.

 

 

> 

 

 

 

 

Download 565_Dec_27_b.mw


The graphs don't quite sow it all, but:

Im(q(x))= Re(q(x)) at 1/4 +1*n | n is an integer!

 

 

 

-

-

Some Numerical Qualities Part 3 Today, we will compare the MRB Constant to a couple of more widely known Constants.

 

 

 

> 

g1:=n->(-1)^n*(n^(1/n)-1):
> 

m:=sum(g1(n),n=1..infinity):evalf(%);
#The MRB Constant

.1878596425

> 

f:=n->(-1)^n*1/(2*n+1)^2:
> 

c:=sum(f(n),n=0..infinity);evalf(%);
#Catalan's Constant

c := hypergeom([1/2, 1/2, 1], [3/2, 3/2], -1)

.9159655942

> 

t1:=exp(arctan(1/2))/m;

t1 := exp(arctan(1/2))/(sum((-1)^n*(n^(1/n)-1), n = 1 .. infinity))

> 

t2:=GAMMA(1/3)*c^2/(10);

t2 := (1/15)*(Pi*3^(1/2)*hypergeom([1/2, 1/2, 1], [3/2, 3/2], -1)^2/GAMMA(2/3))

> 

s1:=evalf(t1*t2,12);

s1 := 1.90216058224

> 

 

> 

s2:=1.90216058304:#Bruns Constant see

http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0208&L=nmbrthry&F=&S=&P=1968)

> 

s1-s2;

-0.1e-8



Could there actually be a connection between the MRB Constant and Bruns' Constant?

> 

 


Download 565_Dec_28.mw

 

This post was generated using the MaplePrimes File Manager

View 565_Dec 28.mw on MapleNet or Download 565_Dec 28.mw
View file details

marvinrayburns.com
searchme!
Marvin Ray Burns Download 565_Dec 28.mw
View file details

-

-

Some Numerical Qualities Part 4 In the first post on this blog, we introduced some series' that share a sequence of partial sums. In this post we will investigate a particular numerical relationship between the sums of these series'. The emphasis is on a numerical relationship; however, we will use plots to represent a large set {} of series'.



> 

 mit := proc (it) global git, m6; git := proc (n) options operator, arrow; (-1)^n*(n^(1/n)-it) end proc; m6 := sum(git(n),n = 1 .. infinity) end proc:

Here, mit(1) is the number we recognize as the MRB Constant.

> 

evalf(mit(1));

.1878596425

In the spirit of the first message of this blog, mit(n) are all family members -- whose series' share partial sums with mit(1).
mit(1):=x->sum((-1)^n*(n^(1/n)-1),n=1..infinity);
mit(2):=x->sum((-1)^n*(n^(1/n)-2),n=1..infinity);
mit(3):=x->sum((-1)^n*(n^(1/n)-3),n=1..infinity);
 and so
mit(c):=x->sum((-1)^n*(n^(1/n)-c),n=1..infinity);.

> 

f:=x->mit(1)/mit(x):

Below,{} means "the set of."
Notice that the ratio of {all real family members}to the mrb constant assumes the shape of an imperfect hyperbola -- roughly between y=0.4/x and y=0.5/x.
Members with 5

> 

extra1:=0.4/x:extra2:=0.5/x:extra3:=0.41/x:
> 

 

> 

plot([extra1,f(x),extra2],x=1..100,color=[green,red,blue]);plot([extra1,f(x),extra2],x=1..100,y=0..0.2,color=[green,red,blue]);plot([extra1,f(x),extra3],x=1..100,y=0..0.2,color=[green,red,blue]);

> 

 

Remember, in Some Plot Qualities Part 3, with complex graphs of the partial sums of f1 [here represented by mit (1)],
we opened up the door for great opportunity for further exploration with elliptical fields.
Now in the ratios of {all real family members}
we opened up the door for great opportunity for further exploration with hyperbolic fields.

> 

 

 



Download 565_dec29b.mw


-

-

Some Numerical Qualities Part 5 This post is an immediate follow-up on Some Numeric Qualities Part 4. In part 4 we found out the ratio of the sum of family members of m to m gave us a hyperbolic field. In this part we will look at the ratio of one family member to the next one. i.e. like As n=1..infinity: sum((-1)^n*n^(1/n-1))/sum((-1)^n*n^(1/n-2)), sum((-1)^n*n^(1/n-2))/sum((-1)^n*n^(1/n-3)), sum((-1)^n*n^(1/n-3))/sum((-1)^n*n^(1/n-4)), etc.



> 

mit := proc (it) global git, m; git := proc (n) options operator, arrow; (-1)^n*(n^(1/n)-it) end proc; m := sum(git(n),n = 1 .. infinity) end proc;

mit := proc (it) global git, m; git := proc (n) options operator, arrow; (-1)^n*(n^(1/n)-it) end proc; m := sum(git(n), n = 1 .. infinity) end proc

In our last post, we discovered the ratio of mit(1)/mit(x) is nearly hyperbolic.

It is not suprizing from the last post, the graph of the function mit, for 4

> 

plot([mit(x),0.5*x],x=1..20,color=[green,red,blue]);
plot([mit(x),0.5*x],x=21..100,color=[green,red,blue]);
plot([mit(x),0.5*x],x=101..200,color=[green,red,blue]);

> 

roll:=evalf(rand(0..100)/100):

Speaking of slope, lets examine the slope of mit:

In the following lists, the step or delta x comes from a multiple of or roll or r.

Working with random steps gives us a chance to find any hidden singularities and occasionally an inability of Maple to do the job we ask.
This gives us a clue about mit(x) for small x::real:

> 

lprint(seq(evalf(mit(x/(300*r))-mit(x/(300*r)+1),10),x=2..10));

-.5000000002, -.5000000000, -.4999999998, -.5000000001, -.4999999999, -.\
5000000003, -.5000000001, -.4999999999, -.5000000002

> 

for a from 1 to 5 do r:=roll():printf("%q",seq(evalf(mit(x/(300*r))-mit(x/(300*r)+1)),x=2..10)) od;

r := .3200000000

-.4999999999, -.5000000000, -.5000000002, -.4999999999, -.5000000000, -.5000000002, -.4999999999, -.5000000000, -.2600570242-1.*sum((-1.)^n*(n^(1/n)-1.104166667),n = 1 .. infinity)

r := .7100000000

-.4999999999, -.5000000000, -.5000000002, -.4999999998, -.5000000000, -.5000000002, -.4999999998, -.5000000000, -.5000000001

r := 0.6000000000e-1

-.2565848020-1.*sum((-1.)^n*(n^(1/n)-1.111111111),n = 1 .. infinity), -.2288070242-1.*sum((-1.)^n*(n^(1/n)-1.166666667),n = 1 .. infinity), -.2010292464-1.*sum((-1.)^n*(n^(1/n)-1.222222222),n = 1 .. infinity), -.1732514686-1.*sum((-1.)^n*(n^(1/n)-1.277777778),n = 1 .. infinity), -.1454736909-1.*sum((-1.)^n*(n^(1/n)-1.333333333),n = 1 .. infinity), -.1176959130-1.*sum((-1.)^n*(n^(1/n)-1.388888889),n = 1 .. infinity), -.4999999999, -.5000000000, -.5000000002

r := 0.8000000000e-1

-.4999999999, -.2496403575-1.*sum((-1.)^n*(n^(1/n)-1.125000000),n = 1 .. infinity), -.2288070242-1.*sum((-1.)^n*(n^(1/n)-1.166666667),n = 1 .. infinity), -.2079736909-1.*sum((-1.)^n*(n^(1/n)-1.208333333),n = 1 .. infinity), -.1871403575-1.*sum((-1.)^n*(n^(1/n)-1.250000000),n = 1 .. infinity), -.1663070242-1.*sum((-1.)^n*(n^(1/n)-1.291666667),n = 1 .. infinity), -.1454736909-1.*sum((-1.)^n*(n^(1/n)-1.333333333),n = 1 .. infinity), -.1246403575-1.*sum((-1.)^n*(n^(1/n)-1.375000000),n = 1 .. infinity), -.5000000002

r := .9700000000

-.4999999999, -.4999999999, -.4999999998, -.5000000002, -.5000000002, -.5000000001, -.5000000001, -.5000000000, -.5000000000

For small enough x, the absolute value of the slope of mit(x),x::real:  tends to be less than 1/2.

 

 

This gives us a clue about the slope of  mit(x) for  not as small x::real: (the x's are around 90,000 times larger than the previous routine.)

> 

for a from 1 to 5 do r:=roll():printf("%q",seq(evalf(mit(300*x/r)-mit(300*x/r+1)),x=2..10)) od;

r := 0.

0., 0., 0., 0., 0., 0., 0., 0., 0.

r := .6600000000

-.5000000, -.5000000, -.5000000, -.500000, -.500000, -.500000, -.500000, -.500000, -.500000

r := .8900000000

-.5000000, -.5000000, -.5000000, -.5000000, -.500000, -.500000, -.500000, -.500000, -.500000

r := .4600000000

-.5000000, -.5000000, -.500000, -.500000, -.500000, -.500000, -.500000, -.500000, -.500000

r := .8300000000

-.5000000, -.5000000, -.5000000, -.5000000, -.500000, -.500000, -.500000, -.500000, -.500000

> 

 

As the size of x increases, the absolute value of the slope of mit(x) for x::real also tends to slightly increase toward  a true 1/2.

  



Download 565_dec_30.mw


-

Some Numerical Qualities Part 6 In Some Numeric Qualities part 4 and 5 we investigated a couple of unique qualities of mit. In this post, we will investigate a theoretically useful quality that is not so unique.

 

> 

g:=x->(-1)^x*(x^(1/x)):
g1:=x->(-1)^x*(x^(1/x)-1):

Previously we defined a process called mit.
Mit yields all of the family member functions, that shares every other partial sum with g, and thus can be used in producing a decimal expansion of the MRB Constant.

This is important for understanding the following, the MRB Constant=mit(1)=g1, mit(0)=g0,  mit(2)=g(2), etc.

> 

restart:
> 

mit := proc (it) global git, m; git := proc (n); (-1)^n*(n^(1/n)-it) end proc; m := sum(git(n),n = 1 .. infinity) end proc;

mit := proc (it) global git, m; git := proc (n) (-1)^n*(n^(1/n)-it) end proc; m := sum(git(n), n = 1 .. infinity) end proc

We know that the MRB Constant is represented by the strategic stacking of hypercubes according to Q1-1.doc; it is a must read.
Additionally, Q1-1.doc has a link to a third part that might be of some interest.
http://www.mapleprimes.com/files/565_QA-1.doc

Now that you have read QA-1.doc, We can discuss the following brief issue.
Numerically, the MRB Constant is the alternating sum of the integers to their own roots.
It can have the form of mit(0), mit(1),...mit(n) or even mit(c), where c is a complex constant

Since, the MRB Constant is the alternating sum of the integers to the power of their own roots, it seems reasonable to ask the following two questions.

 i.    What real numbers to their own roots equal mit(1)?

ii.    What real number do we get when we raise mit(1) to the power of it self an infinite number of times?

The following is not unique to the MRB Constant; however, it only works for a limited range of numbers in which range the MRB lies.

 i.

> 

r:=fsolve(x^(1/x)=mit(1),x);

r := .4619214402

ii.

> 

 

> 

m:=mit(1):
  printf("fsolve(x^(1/x)=mit(1),x)->x=%a and m^(m^(m^(...)))approx.=%a",r,
evalf(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^(m^m)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))));

fsolve(x^(1/x)=mit(1),x)->x=.4619214402 and m^(m^(m^(...)))approx.=.4619214401

 i.    What real numbers to their own roots equal mit(1)?

ii.    What real number do we get when we raise mit(1) to the power of it self an ininate number of times?

i. and ii. have the same answer.

> 

 

 

 

 

Download 565_Dec_31.mw

marvinrayburns.com
searchme!
Marvin Ray Burns Download 565_Dec 31.mw
View file details

-

-

Reurn to the original MRB Constant Blog
infinity

Please Wait...