# 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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