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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