# 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?