Tetration extension for bases between 1 and eta

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Tetration extension for bases between 1 and eta

bo198214 $Offline$
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11/07/2009, 05:11 PM

(11/07/2009, 09:31 AM)bo198214 Wrote: 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?

Actually I think one can show that $\log_b^{\circ k} g_k(x+k)$ is strictly increasing with $k$ because $\log_b g_k(x+k)$ is concave and hence
$g_{k-1}(x+k-1) < \log_b g_k(x+k)$ and thatswhy
$\log_b^{\circ k-1} g_{k-1}(x+k-1)<\log_b^{\circ k} g_{k}(x+k)$
also it is bounded and hence must have a limit.

But now there is still the question why the limit $f(x)$ indeed satisfies
$f(x+1)=b^{f(x)}$ ?

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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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