• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 tiny q: superroots of real numbers x>e Gottfried Ultimate Fellow Posts: 767 Threads: 119 Joined: Aug 2007 02/02/2009, 06:54 PM Hi folks - I've just asked this question in news:sci.math; it is a tiny question and possibly answered anywhere here around ( I didn't follow the superroot-discussion intensely) so maybe we have a link already... Ok, let's go: Let's define the n'th iterative root ("srt") via Code:`f(x,1) = x    f(x,2) = x^x   f(x,3) = x^(x^x)     f(x,k) = ...`as one inverse of f, returning a base if a number and a iteration-count is given, such that, for instance Code:`srt(y,3) = x  --> f(x,3) = y`and consider the sequence Code:`srt(3,1) , srt(3,2), srt(3,3),..., srt(3,k),...  (for k=1 ... inf )` Then: what is x in Code:`x =  lim {k->inf} srt(3,k)` The sequence decreases from 3 down to e^(1/e) + eps but I think, it cannot fall below. Code:```k       x=srt(3,k) --------------------- 1    3.000000     =srt(3,1) 2    1.825455 4    1.563628 8    1.484080 16    1.457948 32    1.449171 64    1.446164 128    1.445135    =srt(3,128) ... ->inf   -> ??       srt(3,inf) ================================     compare other limits inf    1.444668    =e^(1/e) --------------------------------     inf    1.442250    =3^(1/3)``` On the other hand, it should arrive at 3^(1/3)... Do I actually overlook something and the sequence can indeed cross e^(1/e)? Gottfried Gottfried Helms, Kassel « Next Oldest | Next Newest »

 Messages In This Thread tiny q: superroots of real numbers x>e - by Gottfried - 02/02/2009, 06:54 PM RE: tiny q: superroots of real numbers x>e - by bo198214 - 02/02/2009, 10:21 PM RE: tiny q: superroots of real numbers x>e - by Gottfried - 02/03/2009, 12:15 AM RE: tiny q: superroots of real numbers x>e - by bo198214 - 02/03/2009, 10:01 AM RE: tiny q: superroots of real numbers x>e - by Gottfried - 02/03/2009, 11:36 AM RE: tiny q: superroots of real numbers x>e - by bo198214 - 02/03/2009, 12:46 PM

 Possibly Related Threads... Thread Author Replies Views Last Post Green Eggs and HAM: Tetration for ALL bases, real and complex, now possible? mike3 27 39,317 07/02/2014, 10:13 PM Last Post: tommy1729 tiny limit-curiosity [ from ratio b^^(2j) / b^^j ] Gottfried 6 14,386 03/28/2014, 12:35 AM Last Post: tommy1729 Tracing real values of x^(1/x) Gottfried 0 3,386 09/06/2007, 04:33 PM Last Post: Gottfried

Users browsing this thread: 1 Guest(s)