September the 4, 2011, I first described the convergents constants (cc), as rudimentary defined at https://oeis.org/wiki/Convergents_constant . The cc for Floor[ n1 ] < n1 < Ceiling[n1] is the same for any Floor[n1]< nx <Ceiling[n1] , where the n's are rational. Perhaps a little clearer is the example, cc(11/10)=cc(19/10), and cc(21/10)=cc(29/10), but cc(19/10) does not equal cc(21/10) because 1< 19/10< 2 and 2< 21/10< 3.
The cc of a positive integer is that same integer.
Here is the code for convergents constants of some non-integer, rational numbers.
Quiet[Table[ (*Let l start out as a rational non-integer.*)l = Rationalize[x + RandomReal[], 0];
For[a = 1, a < 120, (*make a list of convergents of l.*) c = Convergents[l, 90];
(*Find the continued fraction that has that list as its divisors*) l = FromContinuedFraction[c],
a++];
Print[N[l, 20]], {x, 0, 120}]; // TableForm]
The list of cc's that you get indicate that as n goes to Infinity, cc(n) goes to n. That is interesting, but not too surprising.
As far as cc(10^(nx)) going to 10^(nx), there is a series for that!
Table[x = 10^n; l = x + RandomReal[];
For[a = 1, a < 60, c = Convergents[l, 90];
l = FromContinuedFraction[c], a++]; Print[N[l, 100]], {n, 0, 10}]
gives the a list of cc(10^(nx)) for several x's.
I noticed the short runs of repeating numbers and found a partial pattern to them!
Let
f[n_] := -(10^n) +
Sum[(-1)^(m + 1)*(m + 1)!*10^(-(2 m + 1) n), {m, 0, 3}] -
112*10^(-9*n) + 566*10^(-11*n) - 3000*10^(-13*n) +
16444*10^(-15*n) + 24*10^(-16*n) - 92424*10^(-17 n) -
276*10^(-18*n) + 530320*10^(-19*n) + 3792*10^(-20 *n)
+...,
where (...) follows a pattern that is hard to describe exactly, and then cc(10^nx)+f[n sub x] gives a number magnitudes closer to 0 than that of cc(10^nx ) -10^nx, especially asnx gets any decent size to it at all!
Here is a code with the part of the pattern I found:
Table[x = 10^n;
l = x + Rationalize[RandomReal[WorkingPrecision -> 200]];
For[a = 1, a < 80, c = Convergents[l, 110];
l = FromContinuedFraction[c], a++];
Print[N[l + f[n], 95]], {n, 0, 20}]
which give the following minuscule results:
455353.49999999999999999999999999999999999922805342537191251034213031344782008177019821150735291
2.9199474026688371803570441462414026209663468193828590616799286250429138833291268819435693003382*10^-15
3.0883808365393805976599772401114132776432990139760009925703183747698139661120893356138812636393*10^-36
3.0899412769474228771522360310260164263518211126255365150613157020729793140135461753747853530079*10^-57
3.0899345724509652998138261484536342957238257605233238454711468532965115376145139371649735930125*10^-78
3.0899322736925121154875800561960032313237692698561201626448050392553755207902175257300035940193*10^-99
3.0899320275337251237145063579917678945367126519803736558951266036718173248409899610281152646934*10^-120
3.0899320027550172512397144082357990376848798700608203940718841807568050176487965173555734741126*10^-141
3.0899320002755181725123997143984235799023768547332033956375913675336560958207506367451033410558*10^-162
3.0899320000275519817251239997143974423579902237685532665367291193146563568180419131049229764317*10^-183
3.0899320000027551998172512399997143973442357990222376855385998700624674870208317163285667526111*10^-204
3.0899320000002755199981725123999997143973344235799022223768553919332033958023042576351319781005*10^-225
3.0899320000000275519999817251239999997143973334423579902222237685539252665367291357859813190328*10^-246
3.0899320000000027551999998172512399999997143973333442357990222222376855392585998700624691341537*10^-267
3.0899320000000002755199999981725123999999997143973333344235799022222223768553925919332033958025*10^-288
3.0899320000000000275519999999817251239999999997143973333334423579902222222237685539259252665367*10^-309
3.0899320000000000027551999999998172512399999999997143973333333442357990222222222376855392592586*10^-330
3.0899320000000000002755199999999981725123999999999997143973333333344235799022222222223768553926*10^-351
3.0899320000000000000275519999999999817251239999999999997143973333333334423579902222222222237686*10^-372
3.0899320000000000000027551999999999998172512399999999999997143973333333333442357990222222222222*10^-393
3.0899320000000000000002755199999999999981725123999999999999997143973333333333344235799022222222*10^-414
There are also patterns in the final continued fractions of cc( n sub x), as described at https://oeis.org/wiki/Tableofconvergents_constants .
