Re: [math-fun] Interesting limit question (sort of spoiler alert)
Nice analysis, George! Yup, Hans named it — the Foias constant. Apparently it was discovered because of a misprint in a copy of a question that originally asked whether the (much simpler) recurrence x_(n+1) = (1 + 1/x_n)^x_n (where x_1 > 0) could possibly approach oo. —Dan Hans Havermann wrote: ----- GH: "f(1) = 1.1874523511" Looks like Foias' constant. https://mathworld.wolfram.com/FoiasConstant.html -----
hihi, all - it is likely about what i wrote at the end - precision loss made my program bogus (but it was still interesting) - it seemed possible that a sufficiently slowly increasing function could get large, because then 1 + 1/f(n) is farther away from 1, so the power might be large again, but i didn't try that more later, chris On 2020-09-04 10:29, Dan Asimov wrote:
Nice analysis, George!
Yup, Hans named it — the Foias constant.
Apparently it was discovered because of a misprint in a copy of a question that originally asked whether the (much simpler) recurrence x_(n+1) = (1 + 1/x_n)^x_n (where x_1 > 0) could possibly approach oo.
—Dan
Hans Havermann wrote: ----- GH: "f(1) = 1.1874523511"
Looks like Foias' constant.
https://mathworld.wolfram.com/FoiasConstant.html -----
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Hans: Thanks for pointing me to Foias's constant. The literature around that explains it nicely. Dan: Thanks for posting an interesting and surprising problem. Christopher: I may have reverse-engineered your 1.84 value. Roundoff error couldn't get the calculation so far astray. Playing around with it, I can replicate your result if you coded an off-by-one error in your exponent, like: f(n+1) = (1 + 1/f(n))^(n+1) All: This is a nice example where listing decimal digits in the OEIS is useful. I didn't think to look there as I generally dislike series that are dependent on a base 10 representation, but I have to admit there is a pragmatic value to it in this sort of case. So maybe I should rethink my prejudice... George http://georgehart.com On 9/4/2020 1:53 PM, christopher landauer wrote:
hihi, all -
it is likely about what i wrote at the end - precision loss made my program bogus (but it was still interesting) -
it seemed possible that a sufficiently slowly increasing function could get large, because then 1 + 1/f(n)
is farther away from 1, so the power might be large again, but i didn't try that
more later,
chris
On 2020-09-04 10:29, Dan Asimov wrote:
Nice analysis, George!
Yup, Hans named it — the Foias constant.
Apparently it was discovered because of a misprint in a copy of a question that originally asked whether the (much simpler) recurrence x_(n+1) = (1 + 1/x_n)^x_n (where x_1 > 0) could possibly approach oo.
—Dan
Hans Havermann wrote: ----- GH: "f(1) = 1.1874523511"
Looks like Foias' constant.
https://mathworld.wolfram.com/FoiasConstant.html -----
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hihi, all (esp. George H.) - i don't think it is a simple off by 1 problem (though of course it could be): when i use n-1 instead of n in the exponent, the program finds the threshold at ~3.684; when i use n+1 instead, it gets a threshold at ~1.028 but i did not check that the behavior is the same (eventually alternating 1 and 2^(n-1) or 2^(n+1), with the threshold determining a boundary between the initial values that go to 1 for even n and the initial values that go to 1 for odd n) also, the precision failures (when 1+1/f(n) == 1) are at different times (earlier 30-47 for n-1 than the 56-57 for n, and later 116-117 for n+1 i also did not check that there is a single threshold that separates the eventual behavior into these two cases so something else is weird in my program; oh, well more later, if i figure it out, chris On 2020-09-04 11:15, George Hart wrote:
Hans: Thanks for pointing me to Foias's constant. The literature around that explains it nicely.
Dan: Thanks for posting an interesting and surprising problem.
Christopher: I may have reverse-engineered your 1.84 value. Roundoff error couldn't get the calculation so far astray. Playing around with it, I can replicate your result if you coded an off-by-one error in your exponent, like: f(n+1) = (1 + 1/f(n))^(n+1)
All: This is a nice example where listing decimal digits in the OEIS is useful. I didn't think to look there as I generally dislike series that are dependent on a base 10 representation, but I have to admit there is a pragmatic value to it in this sort of case. So maybe I should rethink my prejudice...
George http://georgehart.com
On 9/4/2020 1:53 PM, christopher landauer wrote:
hihi, all -
it is likely about what i wrote at the end - precision loss made my program bogus (but it was still interesting) -
it seemed possible that a sufficiently slowly increasing function could get large, because then 1 + 1/f(n)
is farther away from 1, so the power might be large again, but i didn't try that
more later,
chris
On 2020-09-04 10:29, Dan Asimov wrote:
Nice analysis, George!
Yup, Hans named it — the Foias constant.
Apparently it was discovered because of a misprint in a copy of a question that originally asked whether the (much simpler) recurrence x_(n+1) = (1 + 1/x_n)^x_n (where x_1 > 0) could possibly approach oo.
—Dan
Hans Havermann wrote: ----- GH: "f(1) = 1.1874523511"
Looks like Foias' constant.
https://mathworld.wolfram.com/FoiasConstant.html -----
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hihi, all - ok, i found part of the problem; it was my mistaken definition for a variable i ran the same program with what i thought were going to be exponents n-1, n, and n+1 (and without GMP): for n-1, it found the threshold in the interval lo 1.187452351126496719 hi 1.187452351126496941 (this is the right answer for the original problem), for n, it found the threshold in the interval lo 1.842139054296648482 hi 1.842139054296648704 (this is what i reported before), for n+1, it found the threshold in the interval lo 2.380377436256564838 hi 2.380377436256565282, and all of them exhibit the peculiar unique threshold property, which leads me to wonder how far away from n the exponent can be and still have this threshold property so these exponents are each off by 1 (what i named as ``n'' in the program was actually (n+1), as was correctly pointed out earlier by George Hart) - oops more soon, chris On 2020-09-04 11:15, George Hart wrote:
Hans: Thanks for pointing me to Foias's constant. The literature around that explains it nicely.
Dan: Thanks for posting an interesting and surprising problem.
Christopher: I may have reverse-engineered your 1.84 value. Roundoff error couldn't get the calculation so far astray. Playing around with it, I can replicate your result if you coded an off-by-one error in your exponent, like: f(n+1) = (1 + 1/f(n))^(n+1)
All: This is a nice example where listing decimal digits in the OEIS is useful. I didn't think to look there as I generally dislike series that are dependent on a base 10 representation, but I have to admit there is a pragmatic value to it in this sort of case. So maybe I should rethink my prejudice...
George http://georgehart.com
On 9/4/2020 1:53 PM, christopher landauer wrote:
hihi, all -
it is likely about what i wrote at the end - precision loss made my program bogus (but it was still interesting) -
it seemed possible that a sufficiently slowly increasing function could get large, because then 1 + 1/f(n)
is farther away from 1, so the power might be large again, but i didn't try that
more later,
chris
On 2020-09-04 10:29, Dan Asimov wrote:
Nice analysis, George!
Yup, Hans named it — the Foias constant.
Apparently it was discovered because of a misprint in a copy of a question that originally asked whether the (much simpler) recurrence x_(n+1) = (1 + 1/x_n)^x_n (where x_1 > 0) could possibly approach oo.
—Dan
Hans Havermann wrote: ----- GH: "f(1) = 1.1874523511"
Looks like Foias' constant.
https://mathworld.wolfram.com/FoiasConstant.html -----
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Kerry, that is an *extremely* cool and thought-provoking set of images. I propose that the points on the boundary be called "Foias points", and then we can talk about the positive and negative real Foias points. Is anybody other than me slightly freaked out by the fact that the microscopic neighborhoods of the positive and negative real Foias points look so similar? They don't look similar at all when you look at the diagram as a whole. Note that the set of Foias points looks much more like a Julia set than it does like a Mandelbrot set, which makes sense because it is asking about the divergence of a *single* iterated function. I suspect that if we iterated (c + 1/f(n))^n (with an arbitrary constant c replacing 1 in the original), we would get a whole family of Julia sets plus a Mandelbrot analogue. On Sat, Sep 5, 2020 at 5:00 PM christopher landauer <topcycal@gmail.com> wrote:
hihi, all -
ok, i found part of the problem; it was my mistaken definition for a variable
i ran the same program with what i thought were going to be exponents n-1, n, and n+1 (and without GMP):
for n-1, it found the threshold in the interval
lo 1.187452351126496719
hi 1.187452351126496941
(this is the right answer for the original problem),
for n, it found the threshold in the interval
lo 1.842139054296648482
hi 1.842139054296648704
(this is what i reported before),
for n+1, it found the threshold in the interval
lo 2.380377436256564838
hi 2.380377436256565282,
and all of them exhibit the peculiar unique threshold property,
which leads me to wonder how far away from n the exponent can be and still have this threshold property
so these exponents are each off by 1 (what i named as ``n'' in the program was actually (n+1), as was correctly pointed out earlier by George Hart) -
oops
more soon,
chris
On 2020-09-04 11:15, George Hart wrote:
Hans: Thanks for pointing me to Foias's constant. The literature around that explains it nicely.
Dan: Thanks for posting an interesting and surprising problem.
Christopher: I may have reverse-engineered your 1.84 value. Roundoff error couldn't get the calculation so far astray. Playing around with it, I can replicate your result if you coded an off-by-one error in your exponent, like: f(n+1) = (1 + 1/f(n))^(n+1)
All: This is a nice example where listing decimal digits in the OEIS is useful. I didn't think to look there as I generally dislike series that are dependent on a base 10 representation, but I have to admit there is a pragmatic value to it in this sort of case. So maybe I should rethink my prejudice...
George http://georgehart.com
On 9/4/2020 1:53 PM, christopher landauer wrote:
hihi, all -
it is likely about what i wrote at the end - precision loss made my program bogus (but it was still interesting) -
it seemed possible that a sufficiently slowly increasing function could get large, because then 1 + 1/f(n)
is farther away from 1, so the power might be large again, but i didn't try that
more later,
chris
On 2020-09-04 10:29, Dan Asimov wrote:
Nice analysis, George!
Yup, Hans named it — the Foias constant.
Apparently it was discovered because of a misprint in a copy of a question that originally asked whether the (much simpler) recurrence x_(n+1) = (1 + 1/x_n)^x_n (where x_1 > 0) could possibly approach oo.
—Dan
Hans Havermann wrote: ----- GH: "f(1) = 1.1874523511"
Looks like Foias' constant.
https://mathworld.wolfram.com/FoiasConstant.html -----
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On 06/09/2020 01:07, Allan Wechsler wrote:
Note that the set of Foias points looks much more like a Julia set than it does like a Mandelbrot set, which makes sense because it is asking about the divergence of a *single* iterated function. I suspect that if we iterated (c + 1/f(n))^n (with an arbitrary constant c replacing 1 in the original), we would get a whole family of Julia sets plus a Mandelbrot analogue.
If you use a constant whose absolute value is > 1, then an "overlarge" x_n gives you (c + small)^n next, and that will be _large_ rather than _small_, so the nature of the behaviour is different in this case. (If the absolute value is < 1, maybe that's OK?) -- g
On 06/09/2020 11:20, I wrote:
On 06/09/2020 01:07, Allan Wechsler wrote:
Note that the set of Foias points looks much more like a Julia set than it does like a Mandelbrot set, which makes sense because it is asking about the divergence of a *single* iterated function. I suspect that if we iterated (c + 1/f(n))^n (with an arbitrary constant c replacing 1 in the original), we would get a whole family of Julia sets plus a Mandelbrot analogue.
If you use a constant whose absolute value is > 1, then an "overlarge" x_n gives you (c + small)^n next, and that will be _large_ rather than _small_, so the nature of the behaviour is different in this case.
(If the absolute value is < 1, maybe that's OK?)
For c=0.5, we get a region kinda like the one for c=1 but with a less crinkly boundary. No big surprises there. For c=0.5 + 0.3i, we get a similar-ish region but its size is hugely increased, with its northern boundary somewhere around 6i. For c=0.8 + 0.6i ... I don't trust the numerics of what I'm doing; it goes rather unstable, not super-surprisingly. For c=0.79 + 0.59i, we _again_ seem to get instabilities. So I think we've found a place where the boundary of the "Mandelbrot set" pokes _inside_ the unit circle. Moving a bit further in in that vicinity, here's c=0.773+0.57i (which doesn't exhibit such instabilities): https://i.imgur.com/sn6Qk4v.png It seems to be disconnected but _not_ dust-like, which you never get with ordinary Julia sets. -- g
hihi, all - i'm replying to this old message because i haven't kept up with the marvelous and interesting pictures being shown - i did run the same three exponents (n, n+1, n+2, despite my previously incorrect names for them) with GMP up to 4096 bits precision (yah, it's overkill, but i was wondering what would happen) - the same precision failure happens in every case: eventually the value of f(n) gets so large that 1 + 1/f(n) == 1 (in that limited precision), and the series alternates thereafter between 1 and a power of 2 (the same as the exponent used); whether the odd n are the power of 2 or the even n are depends on whether the starting value is more than or less than the threshold (respectively), which appears to increase as the exponent increases it also means i have 4096 bits of the thresholds for n, n+1, and n+2 (i will try n-1 and n+3 later on - integer exponents run much faster); the foias constant is the exponent n case; it means that there are several other constants of potential interest more later, chris On 2020-09-04 11:15, George Hart wrote:
Hans: Thanks for pointing me to Foias's constant. The literature around that explains it nicely.
Dan: Thanks for posting an interesting and surprising problem.
Christopher: I may have reverse-engineered your 1.84 value. Roundoff error couldn't get the calculation so far astray. Playing around with it, I can replicate your result if you coded an off-by-one error in your exponent, like: f(n+1) = (1 + 1/f(n))^(n+1)
All: This is a nice example where listing decimal digits in the OEIS is useful. I didn't think to look there as I generally dislike series that are dependent on a base 10 representation, but I have to admit there is a pragmatic value to it in this sort of case. So maybe I should rethink my prejudice...
George http://georgehart.com
On 9/4/2020 1:53 PM, christopher landauer wrote:
hihi, all -
it is likely about what i wrote at the end - precision loss made my program bogus (but it was still interesting) -
it seemed possible that a sufficiently slowly increasing function could get large, because then 1 + 1/f(n)
is farther away from 1, so the power might be large again, but i didn't try that
more later,
chris
On 2020-09-04 10:29, Dan Asimov wrote:
Nice analysis, George!
Yup, Hans named it — the Foias constant.
Apparently it was discovered because of a misprint in a copy of a question that originally asked whether the (much simpler) recurrence x_(n+1) = (1 + 1/x_n)^x_n (where x_1 > 0) could possibly approach oo.
—Dan
Hans Havermann wrote: ----- GH: "f(1) = 1.1874523511"
Looks like Foias' constant.
https://mathworld.wolfram.com/FoiasConstant.html -----
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participants (5)
-
Allan Wechsler -
christopher landauer -
Dan Asimov -
Gareth McCaughan -
George Hart