Länger, Helmut An elementary proof of the convergence of iterated exponentials. (English) Zbl 0865.26005 Elem. Math. 51, No. 2, 75-77 (1996). The author proves the following known result: For a fixed positive real number \(a\) the sequence \(a_1:=a\), \(a_{n+1}:=a^{a_n}\) \((n\geq 1)\) converges if and only if \(e^{-e}\leq a\leq e^{{1\over e}}\).(Reviewer’s remarks: A large number of references on this topic was collected by R. A. Knoebel [Am. Math. Mon. 88, 235-252 (1981; Zbl 0493.26007)]. For more recent results see G. Bachman [Pac. J. Math. 169, No. 2, 219-233 (1995)] and I. N. Baker and P. J. Rippon [Complex Variables, Theory Appl. 12, No. 1-4, 181-200 (1989; Zbl 0644.30014)] and the references given there). Reviewer: J.Elstrodt (Münster) Cited in 2 Documents MSC: 26A18 Iteration of real functions in one variable 40A05 Convergence and divergence of series and sequences Keywords:fixed point; iterated exponentials Citations:Zbl 0493.26007; Zbl 0644.30014 PDFBibTeX XMLCite \textit{H. Länger}, Elem. Math. 51, No. 2, 75--77 (1996; Zbl 0865.26005) Full Text: EuDML