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 $\lim_{n\to\infty} f(n) = e^{1/e}$ where $f(n) = x$ such that ${}^{n}x = n$ Discussion 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 $x = 1$ $x^x = 2$ $x^{x^x} = 3$ $x^{x^{x^x}} = 4$ and so on... is the sequence under discussion. The conjecture is that the limit of this sequence is $e^{1/e}$, also known as eta ($\eta$). Numerical evidence indicates that this is true, as the solution for x in ${}^{1000}x = 1000$ 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 QED regards tommy1729