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 Infinite Pentation (and x-srt-x) andydude Long Time Fellow Posts: 509 Threads: 44 Joined: Aug 2007 04/12/2009, 08:58 AM So this is my attempt at a proof, and there are a bunch of inequalities, which always confuse me , so let me know if there are any mistakes. Let $\eta = e^{1/e}$ as usual. I will use k (instead of n) to avoid confusion with $\eta$. Lemma 1 (Knoebel). $(a < b)$ iff $({}^{k}a < {}^{k}b)$ for all $a,b,k > 1$. Lemma 2. $\eta < {}^{x}\eta < e$ for all real $x > 1$. Proof. The base-$\eta$ tetrational function is continuous and monotonic? Lemma 3. $b^x > x + 1$ for all positive real $x \ge 5$ and real $b > \eta$. Proof. The function $(x+1)^{1/x} < \eta$ for all $x \ge 5$, thus $\eta^x > x+1$. If $b > \eta > 1$, then $b^x > \eta^x$, so $b^x > x+1$. Lemma 4 (lower bound). For all integer $k \ge 3$, $\eta < a_k = \text{srt}_k(k)$. Proof. Since $k \ge 3$, then obviously $k > e$. Together with lemma (2), this implies that ${}^{k}\eta < e < k$. Substituting $k = {}^{k}(a_k)$ (hypothesis), this can be written as ${}^{k}\eta < {}^{k}(a_k)$ which implies $\eta < a_k$ by lemma (1). Lemma 5 (decreasing). For all integer $k \ge 5$ and $a_k = \text{srt}_k(k)$, $a_{k} > a_{k+1}$. Proof. We have ${}^{k}\left(a_{k}\right) = k$ for all integer $k$ by definition. It follows that $ \begin{tabular}{rl} a_{k+1}^{\left({}^{k}\left(a_{k+1}\right)\right)} & = {}^{k+1}\left(a_{k+1}\right) \\ 1 + {}^{k}\left(a_{k+1}\right) & < {}^{k+1}\left(a_{k+1}\right) \\ 1 + {}^{k}\left(a_{k+1}\right) & < k + 1 \\ {}^{k}\left(a_{k+1}\right) & < k \end{tabular}$ by lemma (3). Thus ${}^{k}\left(a_{k+1}\right) < {}^{k}\left(a_{k}\right)$ which implies $a_{k+1} < a_k$ by lemma (1). Theorem. $\lim_{x\to\infty} \text{srt}_x(x) = \eta$ Proof. $\eta < \text{srt}_{k+1}({k+1}) < \text{srt}_k(k)$ for all $k \ge 5$ by lemma (4) and lemma (5). ... In the limit, the squeeze theorem and completeness should guarantee that the limit exists and converges to $\eta$. Is this right? Andrew Robbins « Next Oldest | Next Newest »

 Messages In This Thread Infinite Pentation (and x-srt-x) - by andydude - 04/11/2009, 09:16 AM RE: Infinite Pentation (and x-srt-x) - by bo198214 - 04/11/2009, 09:31 AM RE: Infinite Pentation (and x-srt-x) - by andydude - 04/11/2009, 07:54 PM RE: Infinite Pentation (and x-srt-x) - by nuninho1980 - 04/11/2009, 12:44 PM RE: Infinite Pentation (and x-srt-x) - by andydude - 04/11/2009, 06:42 PM RE: Infinite Pentation (and x-srt-x) - by nuninho1980 - 04/11/2009, 01:24 PM RE: Infinite Pentation (and x-srt-x) - by andydude - 04/11/2009, 06:30 PM RE: Infinite Pentation (and x-srt-x) - by andydude - 04/11/2009, 06:50 PM RE: Infinite Pentation (and x-srt-x) - by nuninho1980 - 04/11/2009, 09:58 PM RE: Infinite Pentation (and x-srt-x) - by andydude - 04/12/2009, 08:19 AM RE: Infinite Pentation (and x-srt-x) - by andydude - 04/11/2009, 08:11 PM RE: Infinite Pentation (and x-srt-x) - by andydude - 04/12/2009, 08:58 AM RE: Infinite Pentation (and x-srt-x) - by bo198214 - 04/13/2009, 05:01 PM RE: Infinite Pentation (and x-srt-x) - by tommy1729 - 05/03/2009, 10:42 PM RE: Infinite Pentation (and x-srt-x) - by bo198214 - 05/03/2009, 10:51 PM RE: Infinite Pentation (and x-srt-x) - by tommy1729 - 05/03/2009, 10:57 PM RE: Infinite Pentation (and x-srt-x) - by bo198214 - 05/03/2009, 11:20 PM RE: Infinite Pentation (and x-srt-x) - by BenStandeven - 05/04/2009, 08:20 PM RE: Infinite Pentation (and x-srt-x) - by bo198214 - 05/04/2009, 08:57 PM RE: Infinite Pentation (and x-srt-x) - by andydude - 04/13/2009, 05:27 AM RE: Infinite Pentation (and x-srt-x) - by bo198214 - 05/31/2011, 10:29 PM

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