Inspired back and forth

Calculus Level 4

n ~ n ~ n ~ . . . \LARGE \tilde n^{\tilde n^{\tilde n^{.^{.^.}}}}

Find the smallest positive integer n n such that the infinite power tower above diverges, where n ~ = n 1 / n \tilde n= n^{1/n} .

Inspiration .

3 None of the others 4 No such n n exists 2

Otto Bretscher by Otto Bretscher , 65, USA

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1 solution

Andreas Wendler
May 29, 2016

How Euler teaches for congergence the base of the tower must be from the interval [ e − e ; e 1 e ] [e^{-e}; e^{\frac{1}{e}}] For all positive x=n x 1 x x^{\frac{1}{x}} is a member of the interval above since this function has a range [ 1 ; e 1 e ] [1; e^{\frac{1}{e}}] Therefore there isn't any positive integer for which the tower diverges.

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Genau! (+1)

Otto Bretscher - 5 years ago

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