How to Compute Power Towers xxx ...
A power tower (also known as a tetration or iterated exponential) is a mathematical operation that extends the idea of exponentiation. The nth power of x, denoted by xn, means x multiplied by itself n times.The nth iterated exponent of x, usually denoted by
x↑↑n, and less often by
means x exponentiated by itself n times. For instance, x↑↑2 = xx, and x↑↑3 = xxx, etc.
For numbers within a very small range, you can even define an infinite power tower (infinite tetration), in which you exponentiate a number by itself infinitely many times. You can calculate infinte tetration using the calculator on the left. For numbers outside of the given range, the operation does not converge to a single, finite number.
Growth of Power TowersPower Towers grow even more rapidly than exponentiated numbers. Consider the first, second, third, and fourth iterated exponents of 2:
12 = 2
22 = 22 = 4
32 = 222 = 24 = 16
42 = 2222 = 216 = 65536
But 52 = 265536 = is a number with 19729 digits. (You can see it here.)
Infinite Power TowersThe infinite power tower x↑↑∞ (or ∞x) converges so long as
e-e < x < e1/e.
It can be written in closed form using the Lambert W function. The formula is
xxxx ... = -W[-ln(x)]/ln(x)
Some particular values of x↑↑∞ are
1↑↑∞ = 1
√2↑↑∞ = 2
(1/e)↑↑∞ = 0.5671432904, the solution to x = -ln(x)
(1/4)↑↑∞ = 1/2
Zero is an example of a number whose power towers 0↑↑n do not converge as n goes to ∞. The value of 0↑↑n oscillates between 0 and 1 depending on whether n is even or odd
10 = 0↑↑1 = 0
20 = 0↑↑2 = 00 = 1
30 = 0↑↑3 = 000 = 01 = 0
40 = 0↑↑4 = 0000 = 00 = 1
(Note that although 00 is an indeterminate form, in power towers it is defined as 1 since the limit of the function f(x) = xx goes to 1 as x goes to 0.)
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