Tetration towers

Calculus Level 4

Consider the infinitely nested exponential equation:

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

One might naively say... Easy, just substitute in, and:

\(x^N = N\), 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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