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An elementary proof of the convergence of iterated exponentials. (English) Zbl 0865.26005

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).

MSC:

26A18 Iteration of real functions in one variable
40A05 Convergence and divergence of series and sequences
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