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 Tetration extension for bases between 1 and eta bo198214 Administrator Posts: 1,395 Threads: 91 Joined: Aug 2007 11/07/2009, 09:31 AM Let me rephrase in my words: We consider the linear functions $g_k$ on (k-1,k) determined by $g_k(k-1)=f(k-1)={^{ k-1}b}$ and $g_k(k)=f(k)={^k b}$, they are given by: $g_k(x) = ({^k b} - {^{ k-1}}b)(x-k) + {^k b}$. As f is concave $f(x+k) \ge g_k(x+k)$ for $x\in (-1,0)$, and as $\log_b$ is strictly increasing we have also $f(x) \ge \log_b^{\circ k} g_k(x+k)$. On the other hand we know that $f(x+k)=\exp_b^{\circ k}(f(x))\uparrow a$ for $k\to\infty$, hence $a > f(x+k) > g_k(x+k)$ and $g_k(x+k)\uparrow a$ for $k\to\infty$. We have now $f(x+k)-g_k(x+k)\downarrow 0$ for $k\to\infty$. But I think that does not directly show the convergence $f(x) - \log_b^{\circ k} g_k(x+k)\downarrow 0$. Any ideas? « Next Oldest | Next Newest »

 Messages In This Thread Tetration extension for bases between 1 and eta - by dantheman163 - 11/05/2009, 03:00 AM RE: Tetration extension for bases between 1 and eta - by bo198214 - 11/05/2009, 01:44 PM RE: Tetration extension for bases between 1 and eta - by dantheman163 - 11/05/2009, 11:53 PM RE: Tetration extension for bases between 1 and eta - by bo198214 - 11/07/2009, 09:31 AM RE: Tetration extension for bases between 1 and eta - by bo198214 - 11/07/2009, 05:11 PM RE: Tetration extension for bases between 1 and eta - by bo198214 - 11/07/2009, 08:12 PM RE: Tetration extension for bases between 1 and eta - by dantheman163 - 11/07/2009, 11:30 PM RE: Tetration extension for bases between 1 and eta - by bo198214 - 11/08/2009, 02:44 PM RE: Tetration extension for bases between 1 and eta - by mike3 - 11/12/2009, 07:11 PM RE: Tetration extension for bases between 1 and eta - by dantheman163 - 12/15/2009, 01:01 AM RE: Tetration extension for bases between 1 and eta - by bo198214 - 12/15/2009, 01:40 AM RE: Tetration extension for bases between 1 and eta - by dantheman163 - 12/15/2009, 01:48 AM RE: Tetration extension for bases between 1 and eta - by dantheman163 - 12/17/2009, 02:40 AM RE: Tetration extension for bases between 1 and eta - by bo198214 - 12/17/2009, 10:59 AM RE: Tetration extension for bases between 1 and eta - by dantheman163 - 12/19/2009, 05:06 AM RE: Tetration extension for bases between 1 and eta - by bo198214 - 12/19/2009, 10:55 AM

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