\(\displaystyle \sqrt{2}, {\sqrt{2}^\sqrt{2}},{ \sqrt{2}^{\sqrt{2}^\sqrt{2}}}, \cdots \)
is de rij \(1.41421\cdots, 1.63252\cdots, 1.76083\cdots, \cdots\)
Deze rij convergeert naar\(\displaystyle \sqrt{2}^{\sqrt{2}^{\sqrt{2}^{.^{.^{.}}}}} = 2\)
Algemener:
Stel
\(\displaystyle x^{x^{x^{.^{.^{.}}}}} = a\)
dan is\(\displaystyle x^{\left(x^{x^{x^{.^{.^{.}}}}}\right)} = x^a = a\)
dus
\(x = \sqrt[a]{a}\)
Even kijken of ik het goed heb gedaan.
Neem a = 2, dan is
\(x = \sqrt{2}\)
.Correct.
Neem nu a = 4; dan is
\(x = \sqrt[4]{4} = \sqrt{2}\)
Hé, dat is vreemd, want nu is \(\displaystyle 4 = \sqrt{2}^{\sqrt{2}^{\sqrt{2}^{.^{.^{.}}}}} = 2\)
Rara politiepet.
