power tower sequence

For positive values of $a$, the power tower sequence

$$a,a^a,a^{a^a},a^{a^{a^a}},\mathrm{\dots }$$

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 root (http://planetmath.org/Equation) of the equation

$$a^x=x.$$

The proof is found in [1].

Title	power tower sequence
Canonical name	PowerTowerSequence
Date of creation	2013-03-22 16:41:58
Last modified on	2013-03-22 16:41:58
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	8
Author	pahio (2872)
Entry type	Example
Classification	msc 40-00
Related topic	PowerFunction
Related topic	OrderOfOperations
Related topic	NaturalLogBase
Related topic	FunctionXx
Related topic	SuperexponentiationIsNotElementary
Related topic	ErnstLindelof
Defines	power tower sequence