Power Tower

Calculus Level 5

Let h ( x ) = x x x … h(x)=x^{x^{x^{\ldots }}} ; if [ a , b ] [a,b] is the largest domain of positive real values of x x for which h ( x ) h(x) converges, what is the value of a + b a+b (to 3 decimal places)?


The answer is 1.511.

Miles Koumouris by Miles Koumouris , 20, Australia

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1 solution

Marco Brezzi
Aug 1, 2017

The infinite tetration of x x coverges if and only if

x ∈ [ e − e , e 1 e ] x\in\left[e^{-e},e^{\frac{1}{e}}\right]

Making the answer

e − e + e 1 e ≈ 1.511 e^{-e}+e^{\frac{1}{e}}≈\boxed{1.511}

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