06/11/2011, 02:33 PM
Hi James,
I do not really know, whether the following matches your input here; but screening through older discussions I just found an older post of Mike (I'd saved it by copying from google.groups). He observed the following and asked
 & \approx& 1.09861228867 \\<br />
t_2 &=& \log(\log(3^3)) & \approx& 1.19266011628 \\<br />
t_3 &=& \log(\log(\log(3^{3^3}))) & \approx& 1.22079590713 \\ <br />
\vspace8 & & \\ <br />
\vspace8 ... &=& ... \\ <br />
\vspace8 & & \\ <br />
t_{n\to \infty} &\to & \text{constant (which?)} & \approx& 1.22172930187 <br />
\end{eqnarray}<br />
)
Henryk had answered with some proof of convergence and rate of convergence. I had an idea to reformulate this in a way using somehow "lower degree operators" than addition but could not make it better computable, so I didn't involve then further.
If I get your approach right this can be used for such "lower order" operators? Say
 \\<br />
a &+_{\tiny -2} & b &=& \log(\log(e^e^a + e^e^b)) \\<br />
\vspace8 \end{eqnarray} )
and )
for the h-fold iterated log( 3) then Mike's limit can be expressed
 \\<br />
t_2 &=& L^{\tiny o 1}(3) &+& L^{\tiny o 2}(3) \\<br />
t_3 &=& L^{\tiny o 1}(3) &+& L^{\tiny o 2}(3) &+_{\tiny -1}& L^{\tiny o 3}(3) \\<br />
<br />
t_4 &=& L^{\tiny o 1}(3) &+& L^{\tiny o 2}(3) &+_{\tiny -1}& L^{\tiny o 3}(3) &+_{\tiny -2}& L^{\tiny o 4}(3) \\<br />
\vspace8 & &\\<br />
...& &... \\ <br />
\vspace8 & &\\<br />
t_{n\to \infty} &\to & \text{constant} \\<br />
\end{eqnarray} )
where the operator-precedence is lower the more negative the index at the plus is (so we evaluate it from the left).
First question: is this in fact an application of your "rational operator"?
And if it is so, then second question: does this help to evaluate this to higher depth of iteration than we can do it when we try it just by log and exp alone (we can do it to iteration 4 or 5 at max I think) ?
Gottfried
cite:
I do not really know, whether the following matches your input here; but screening through older discussions I just found an older post of Mike (I'd saved it by copying from google.groups). He observed the following and asked
Henryk had answered with some proof of convergence and rate of convergence. I had an idea to reformulate this in a way using somehow "lower degree operators" than addition but could not make it better computable, so I didn't involve then further.
If I get your approach right this can be used for such "lower order" operators? Say
and
where the operator-precedence is lower the more negative the index at the plus is (so we evaluate it from the left).
First question: is this in fact an application of your "rational operator"?
And if it is so, then second question: does this help to evaluate this to higher depth of iteration than we can do it when we try it just by log and exp alone (we can do it to iteration 4 or 5 at max I think) ?
Gottfried
cite:
Quote:In article
<20f2d3ea-1e87-45dc-86b2-a917f89e9370@i18g2000pro.googlegroups.com>,
mike3 <mike4ty4@yahoo.com> wrote:
> Hi.
>
> I noticed this.
>
> log(3) ~ 1.098612288668109691395245237
> log(log(3^3)) ~ 1.192660116284808707569579569
> log(log(log(3^3^3))) ~ 1.220795907132767865324020633
> log(log(log(log(3^3^3^3)))) ~ 1.221729301870251716316203810
> (calculated indirectly via identity log(x^y) = y log(x).)
> log(log(log(log(log(3^3^3^3^3))))) ~ 1.221729301870251827504003124
> (calculated indirectly via identity log(log(x^x^y)) = y log(x) + log
> (log(x)).)
>
> It seems to be stabilizing on some weird value, around 1.2217293.
> What is this? And we seem to run out of log identities here making
> it infeasible to compute further approximations.
>
> Has this been examined before?
Gottfried Helms, Kassel