Look at the function defined by an `infinite tower’ of exponents:

It would seem that for *x*>1 this must blow up. But, amazingly, this is not so.

In fact, the function has finite values for positive *x* up to . We call this function the *power tower function*.

We can construct the power tower function by an iterative process:

This generates the infinite sequence of successive approximations

We note the convention

Thus, the tower is constructed *downwards*; it should really be denoted as

as each subsequent *x* is adjoined to the *bottom* of the tower.

If the sequence converges to *y*, it follows that

This gives an** explicit expression for ***x* as a function of *y*:

Clearly, this function is well defined for all positive *y*. If we can invert it, we can get *y* as a function of *x*.

** Iterative Solution **

The logarithm of gives or

where . This is in a form suited for iterative solution.

Given a value of , and therefore of , we seek a value such that the graph of intersects the diagonal line . Starting from some value , we compute the iterations

In the figure below, we sketch the graph of for a selection of values of . For there is a single root. For , there are two roots. For there is one double root. Finally, for there are no roots.

We plot the function in the figure below. It is defined for all positive . Its derivative vanishes at where it takes its maximum value .

The function is monotone increasing on the interval and has an inverse function on this interval. This inverse is the power tower function, plotted in the figure below.

**Conclusion **

The power tower function is well defined on the domain . This is surprising, as a cursory glance would suggest divergence for .

It would be interesting to investigate the behaviour of the power tower function for complex values of the argument.

A follow-up post, relating the power tower function to the Lambert W-function.

A brief note with more technical details is here ( PDF ).

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