I came across infinite tetrations on wikipedia (https://en.wikipedia.org/wiki/Tetration) it says that the infinite tetration converges if and only if $\ e^{-e} \leq x \leq e^{1/e}$. I was wondering if there is a proof of this.
[Math] Infinite tetration convergence
convergence-divergenceexponentiationreal-analysistetration
Related Solutions
First, I would recommend reading "Exponentials Reiterated" by R.A.Knoebel, http://mathdl.maa.org/mathDL/22/?pa=content&sa=viewDocument&nodeId=3087&pf=1. It is by far the most comprehensive resource on infinite tetration. Second, your question might have been answered sooner if you had posted on the Tetration Forum, http://math.eretrandre.org/tetrationforum/index.php. Lastly, I think it is very encouraging to see others interested in this subject, since I've been interested in it for quite a long time now.
This is base2 Tetration, using Kneser's method, which is analytic in the upper and lower halves of the complex plane. The conjecture is that Kneser's solution is the only one with derivative>0 for z>-2, and which is analytic in the upper and lower halves of the complex plane, and at the real axis for z>-2. It has singularities at negative integers<=-2. Below is the Taylor series evaluated at z=0. The graph below goes from -1.9 to 3. Tet2(3.49090471067047)~=100.
{tet2= 1
+x^ 1* 0.889364954620976
+x^ 2* 0.00867654896536993
+x^ 3* 0.0952388000751818
+x^ 4* -0.00575234854012612
+x^ 5* 0.0129665820200372
+x^ 6* -0.00219604962303099
+x^ 7* 0.00199674684791144
+x^ 8* -0.000563354814878522
+x^ 9* 0.000348242328188164
+x^10* -0.000128532441264720
+x^11* 0.0000670819244205308
+x^12* -0.0000282987528227980
+x^13* 0.0000138001319906329
+x^14* -0.00000620190939837452
+x^15* 0.00000295556146480966
+x^16* -0.00000136867922453470
+x^17* 0.000000649057075651896
+x^18* -0.000000305166939328926
+x^19* 0.000000144948206151230
+x^20* -0.0000000687466431137918
+x^21* 0.0000000327674451778934
+x^22* -0.0000000156310874679970
+x^23* 0.00000000747812838918081
+x^24* -0.00000000358267681323948
+x^25* 0.00000000171986774579521
+x^26* -0.000000000826815962468010
+x^27* 0.000000000398110468382690
+x^28* -1.91942992587732 E-10
+x^29* 9.26631867862980 E-11
+x^30* -4.47869882913349 E-11
+x^31* 2.16712032119157 E-11
+x^32* -1.04969742914240 E-11
+x^33* 5.08944818189222 E-12
+x^34* -2.46987831946185 E-12
+x^35* 1.19965565647417 E-12
+x^36* -5.83165889022053 E-13
+x^37* 2.83702361814325 E-13
+x^38* -1.38118252499282 E-13
+x^39* 6.72883824735072 E-14
+x^40* -3.28030864119803 E-14
+x^41* 1.60015058245703 E-14
+x^42* -7.81025880698438 E-15
+x^43* 3.81431246327820 E-15
+x^44* -1.86381177420546 E-15
+x^45* 9.11196869462329 E-16
+x^46* -4.45694121263769 E-16
+x^47* 2.18105634006697 E-16
+x^48* -1.06780883356354 E-16
+x^49* 5.23008408468722 E-17
+x^50* -2.56274120196474 E-17
}
Best Answer
This relates to the deeper questions of where does the iterates of $a^z$ converge in the complex plane for infinite exponential towers at a fixed point $A$ such that $a^A=A$ and where is the onset for period $n$ behavior.
Let $\Delta z$ be an infinitesimal. Since $a^A=A$, $a^{A+\Delta z}=A a^{\Delta z}=A + Ln A \ \! {\Delta z}$. Therefore $A+\Delta z \Rightarrow A + Ln A \ \! {\Delta z}$. The dynamics of $a^z$ in the neighborhood of a fixed point $A$ are solely dependant on the location of $A$.
Let $Ln A=e^{2 \pi i}$, then $A =e^{e^{2 \pi i}}$. But $a=A^{1/A}$, so $ a=\LARGE e^{e^{2 z i \pi-e^{2 z i \pi}}}$. Let $z=1$, then so $1.444\approx e^{1/e} = \LARGE e^{e^{2 i \pi-e^{2 i \pi}}}$. Let $z=1/2$, then so $0.065988 \approx e^{-e} = \LARGE e^{e^{i \pi-e^{i \pi}}}$
The complex function $\LARGE e^{e^{2 z i \pi-e^{2 z i \pi}}}$
The exponential Mandelbrot set by period. Red is the area of convergence or period 1, orange is period 2 and includes 0, yellow is period 3.