40-00-PowerTowerSequence.tex

\documentclass[12pt]{article}
\usepackage{pmmeta}
\pmcanonicalname{PowerTowerSequence}
\pmcreated{2013-03-22 16:41:58}
\pmmodified{2013-03-22 16:41:58}
\pmowner{pahio}{2872}
\pmmodifier{pahio}{2872}
\pmtitle{power tower sequence}
\pmrecord{8}{38913}
\pmprivacy{1}
\pmauthor{pahio}{2872}
\pmtype{Example}
\pmcomment{trigger rebuild}
\pmclassification{msc}{40-00}
\pmrelated{PowerFunction}
\pmrelated{OrderOfOperations}
\pmrelated{NaturalLogBase}
\pmrelated{FunctionXx}
\pmrelated{SuperexponentiationIsNotElementary}
\pmrelated{ErnstLindelof}
\pmdefines{power tower sequence}

\endmetadata

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\theoremstyle{definition}
\newtheorem*{thmplain}{Theorem}

\begin{document}
For positive values of $a$, the {\em power tower sequence}
              $$a,\, a^a,\, a^{a^a},\, a^{a^{a^a}},\, \ldots$$
is convergent if and only if
               $$\frac{1}{e^e} \leqq a \leqq e^{\frac{1}{e}},$$
approximately 
                  $$0.065989\leqq a \leqq 1.444667.$$
The limit of the sequence is the least real \PMlinkname{root}{Equation} of the equation
                           $$a^x = x.$$
The proof is found in [1].

\begin{thebibliography}{8}
\bibitem{lindelof}{\sc E. Lindel\"of}: {\em Differentiali- ja integralilasku
ja sen sovellutukset III. Toinen osa.}\, Mercatorin Kirjapaino Osakeyhti\"o, Helsinki (1940).
\end{thebibliography}
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\end{document}