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From theFun Factfiles, here is a Fun Fact at the Medium level:

Tower of Powers

Figure 1

Consider an infinite "tower of powers" of x, defined by

x^x^x^... = x^(x^(x^...))

Can we find a value of x so that this tower is equal to 2?
Let's write A for the value of the tower; then also
A = x^{A} from the definition,
so solving for x we get
x = A^{1/A}. If A = 2, then x = Sqrt[2].

How about A = 4? For this one we get

x = 4^{1/4} = sqrt(2),

the same as for A = 2.
In other words, we conclude that the tower of powers

Sqrt(2)^Sqrt(2)^Sqrt(2)... = both 2 and 4.

Which is correct?

Presentation Suggestions:
Present as above and ask the students as homework
to think of explanations for this strange behavior.
At some point they will discover (or need to be shown)
that the tower can be defined as the limit of a sequence:
x, x^x, x^x^x, ...;
they should then try calculating numerically the limit
for x = Sqrt(2) to see if they can decide which answer,
2 or 4, is correct.

The Math Behind the Fact:
Students need some background in limits to understand this,
since the critical point is that the tower has to be defined somehow;
we can't just write down an infinite series of symbols and assume that it has a meaning.
(Euler could do this, but we can't.)
It turns out that the equation has no solutions for A > e,
which has a simple graphical explanation once the iteration is understood.
This example also illustrates the hazards of calculating a "solution" to an equation without knowing that a solution exists.
The article by Mitchelmore has good diagrams and a careful consideration of all the cases.

How to Cite this Page:
Su, Francis E., et al. "Tower of Powers."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.

"A Matter of Definition" by M. C. Mitchelmore.
in A Century of Calculus, Part II edited by Tom Apostol et al., pp. 84-88. Reprinted from
the Amer. Math. Monthly, 81(1974), 643-647.

Keywords: calculus, real analysis, convergence
Subjects: calculus, analysis
Level: Medium
Fun Fact suggested by: Allen Stenger
Suggestions? Use this form.