The hyperexponential function an exercise in convergence

Extract

For positive x, define the sequence of functions {h_n(x)} thus:

So, for example,

x appearing 3 times,

and x appearing 5 times.

The hyperexponential function is then defined as

for those values of x for which this limit exists. Since an infinite number of operations are involved, we clearly need to study carefully the question of convergence.