Infinite Power Towers

Calculus Level 2

Obviously, $1^{1^{1^{.^{.^{.}}}}}=1$

Furthermore, with a little bit of effort, we can show that $\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{.^{.^{.}}}}}=2$

However, clearly $2^{2^{2^{.^{.^{.}}}}}=\infty$

So what is the largest real number $x$ such that $x^{x^{x^{.^{.^{.}}}}}$ is a finite number?

\frac{\pi}{2}

\frac{13}{9}

{\left(\frac{\pi}{e}\right)}^e

e^{\frac{1}{e}}

\pi^{\frac{1}{\pi}}

\sqrt{2}

\sqrt[3]{3}

\frac{3}{2}