Inspired by Otto Bretscher and Abhay Tiwari

Calculus Level pending

\(\huge x^{x^{x^{x^{.^{.^{x^{n}}}}}}}=2\)

The above power tower can be viewed as a recurrence relation: \(a_0=n, a_{k+1}=x^{a_{k}}\) for \( k \ge 0\) and \(\displaystyle\lim_{k \to \infty}a_k=2\).

It is known that \(n\) is a constant and \(x=\sqrt{2}\) is the solution of the above equation. What is the largest range for \(n\)?

Inspiration and more inspiration.


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