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Talk:Convergents constant

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I was trying to figure out a name and I thought Iterated continued fractions from convergents where the process would be explained. Then, since there is actually an infinity of constants, at least one for each interval maybe this first constant's page should be called Iterated continued fractions from convergents' first constant, or something similar. Also, since most of the current contents of Convergents constant is a discussion, it would belong to Talk:Convergents constant (or Talk:Iterated continued fractions from convergents' first constant,) while Convergents constant (or Iterated continued fractions from convergents' first constant) should contain the final results. — Daniel Forgues 23:10, 7 May 2011 (UTC)

Yes, the discussion should be moved. I was hoping, however, that we could come up with a few conclusions (perhaps what we know that we don't know) to post on the page at the same time. — Marvin Ray Burns 01:51, 8 May 2011 (UTC)
As for the name(s), Iterated continued fractions from convergents is more descriptive than this present working one. When we get the page a little more organized, I will ask Steven Finch if he would look at it and see if he has any input as to a name. — Marvin Ray Burns 01:59, 8 May 2011 (UTC)

Convergents constant

If you were to use the first 6 convergents of the MRB constant as terms of a generalized continued fraction, it would give

Using the set of all the convergents of the MRB constant in the same manner,

0.9002582559125551989546840469792811569380853724446300786367882532 has a simple continued fraction expansion

giving rise to the sequence that might be worthy of being entered into the OEIS

{0, 1, 9, 38, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 2, 1, 2, 11, 1, 4, 1, 1, 3, 2, 18, 3, 1, 8,4, 3, 10, 8, 1, 651, 1, 2, 2, 1, 1, 4, 1, 31, 6, 1, 2, 2, 4, 2, 1, 1, 2, 8, 10, 8, 1,2, 1, 1, 1, 125, 5, ...}

Repeat this process of using convergents as terms indefinitely and you get a new constant that is not unique to having started with the MRB constant.

One should try to duplicate this experiment for his or her own self. — Marvin Ray Burns

Using just a finite number of convergents is rather arbitrary,... I would skip it. Using all the convergents might be more interesting, except that I have not yet seen using the convergents of a continued fraction used as terms of a generalized continued fraction, you could just repeat the process ad infinitum... are you saying that doing the process ad infinitum might converge to a new constant? How would you prove that such a process actually converges? How fast would it converge? This seems a lot of work, would the resulting constant, if there is, be interesting? Why? Unless this ad infinitum process would give the same constant, whatever the initial continued fraction (thus whatever the initial constant,) now that would be very interesting! How would you prove such a thing, if it was the case? — Daniel Forgues 01:01, 25 April 2011 (UTC)
You do get that same constant for a wide range of initial values and their continued fractions. I discovered this while writing the post at http://www.mapleprimes.com/posts/102477-A-Generalized-Continued-Fraction ; that should answer some questions. — Marvin Ray Burns
Is this only empirical or is there a proof, and what would be the (provable?) range of initial values? — Daniel Forgues 01:57, 25 April 2011 (UTC)
I wonder how one would prove the range of initial values that give the new constant that I talked about. — Marvin Ray Burns
If you do get that same constant for a wide range of initial values and their simple continued fractions, then that would not be specifically related to the MRB constant and should go on its own page if there is enough known about it. I'm not sure what would be a good short enough title. So I don't think the contents above should be put on the MRB constant page, only a link pointing to that new page for any constant that would fall within that range (or does it depend on some properties or patterns of the simple continued fraction?) — Daniel Forgues 18:27, 25 April 2011 (UTC)
You asked, "Does it depend on some properties or patterns of the simple continued fraction?" The following idea makes computation a little harder. Given n such that the continued fraction of n does not terminate; take the fractional part of n and use it, and its convergents, as we did before; then what do we get? Some sampling does show we often get our "from convergents constant", 0.555753104278045912445405869381... As an example, entering
Table[{N[l = RandomReal[WorkingPrecision -> 100]], 
  Table[c = Convergents[l, 100]; 
  l = FromContinuedFraction[c], {n, 1, 50}];
N[l, Floor[30]]}, {m, 1, 100}] // TableForm
into Mathematica seems to "always?" give that constant, when the initial input>0.04468517[2].
Initial input < 0.04468517[2] seems to be extremely sensitive to variations!!!
Example
l = .04468513845124463; 
Table[c = Convergents[l, 100];
l = FromContinuedFraction[c], {n, 1, 50}]; N[l, 20]
gives 3.2765033850144244631, 

and
 
l = .044685138451244635; 
Table[c = Convergents[l, 100];
l = FromContinuedFraction[c], {n, 1, 50}]; N[l, 20] 
gives 0.55575310427804591245

and

l = .044685138451244635105; 
Table[c = Convergents[l, 100];
l = FromContinuedFraction[c], {n, 1, 50}]; N[l, 20] 
gives 3.2765033850144244631

Marvin Ray Burns 02:08, 28 April 2011 (UTC)