The Repeated Power Function

Leonhard Euler proved that the function , where appears times, approaches a limit for as . (The red points indicate the limits at the endpoints of the interval of convergence.) This Demonstration shows the values of for to . You can zoom in to see the limits at and . The expression bifurcates at the lower endpoint () with . The sequence converges to 1 as from above while converges to 0. Beyond the upper endpoint () the expression rapidly grows to infinity.