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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