Note that this is not the same as R = (((N ^ N) ^ N) ^ N) ...
Then R is the limit of the sequence N, N^N, N^(N^N), ... and is a function of N on a suitable domain. Obviously N = 1 yields R = 1, and we want the maximum N for which R exists, so we need only consider N >= 1 and will assume this from now on. Then if N2 > N1, we must have N2^N2 > N1^N1, N2^(N2^N2) > N1^(N1^N1), and so on; and thus R2 > R1 if they exist. That is, where the function R of N exists, it is monotone increasing. Therefore N as a function of R is also monotone increasing in the range of values of N for which R exists.
We will now examine N as a function of R. Substituting equation (1) into itself, we get
- Taking the logarithm of both sides of (2)
- derivative to zero
- if the second derivative is negative at that point
Thus equation (3) and therefore (2) ceases to yield the monotone increasing behavior that equation (1) does, at the point where R = e and therefore (by (3)) N = e^(1/e). Since (1) was monotone increasing throughout the domain where R exists, this is the maximum N for which R can exist.