Tetration Forum
Another proof of TPID 6 - Printable Version

+- Tetration Forum (https://math.eretrandre.org/tetrationforum)
+-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1)
+--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3)
+--- Thread: Another proof of TPID 6 (/showthread.php?tid=479)

Another proof of TPID 6 - tommy1729 - 07/25/2010

(10/07/2009, 12:03 AM)andydude Wrote: Conjecture

where such that


To evaluate f at real numbers, an extension of tetration is required, but to evaluate f at positive integers, only real-valued exponentiation is needed. Thus the sequence given by the solutions of the equations

and so on... is the sequence under discussion. The conjecture is that the limit of this sequence is , also known as eta (). Numerical evidence indicates that this is true, as the solution for x in is approximately 1.44.

lim n-> oo x^^n = n conj : any real x = eta

since (eta+q) ^^ n grows faster than n for any positive q , we can use the squeeze theorem

lim q -> 0 eta =< x <= eta + q

hence x = eta

see also http://en.wikipedia.org/wiki/Squeeze_theorem