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.
by
Geoff Pilling