Tetration towers

Calculus Level 4

Consider the infinitely nested exponential equation

\[\large x^{x^{x^{\cdot^{\cdot^{\cdot}}}}} = N.\]

One might naively say, "Easy, just substitute in," and

\[x^N = N, \ \text{ so }\ x = \sqrt[N]{N}.\]

However, this doesn't converge for all \(N\). What is the highest \(N\) for which it does?

Give your answer to 3 decimal places.


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