Exact and Unique solution for base e^(1/e)

[bo198214]

But I'm only looking at the first 15 or so terms of the series. It seems pretty well behaved and very well-defined, but maybe I'm missing something?

What exactly did you try?
Did you experiment with the iterations of \( e^x-1 \)? *wondering*

My assertion was that for \( f(x)=e^x-1 \) the convergence radius of the unique formal series \( f^{\circ t}(x) \), (for example given by the double binomial formula) is 0 for every non-integer \( t \). For natural numbers \( t \) it is clear that the radius of convergence is infinity. And for negative numbers \( t=-n \), note that \( f^{-1}(x)=\log(x+1) \) is defined on \( (-1,\infty) \) and has convergence radius 1 when developed at 0, the convergence radius is \( -f^{\circ n-1}(-1) \).