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November 29th, 2017, 04:21 PM   #1
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Solving Square Super-Root with Elementary Functions

First, a little background. I joined to ask this question, so apologies if it is in the wrong sub-forum. I am a senior in high school in Calc 1, but I have recently been intrigued with hyper operations.

With a little experimenting in the calculator and some logic, I came to the conclusion that I can find the square super-root of any real number on the domain (~.6923, ∞) using a chain (not sure if right word) of elementary functions.

Specifically, if we define x^x = y, then:

a = y^(1/y)
b = y^(1/a)
c = y^(1/b)
d = y^(1/c)
...
and so on.

Eventually, the value of y^(1/n) converges to the correct square super-root of y.

If x<1, it asymptotes to the square super-root from above.
If x>1, it oscillates above and below the correct value and eventually converges. However, computing the square super-root of numbers larger than 15 requires carrying a higher number of digits than I currently can in a computer program.

Here is an example. I only wrote down 3 digits each time for brevity, but I carried 10 digits in the calculator.

if x^x = 3:

a = 3^(1/3) = 1.44
b = 3^(1/1.44) = 2.14
c = 3^(1/2.14) = 1.67
d = 3^(1/1.67) = 1.93
e = 3^(1/1.93) = 1.77
f = 3^(1/1.77) = 1.86
g = 3^(1/1.86) = 1.80
h = 3^(1/1.80) = 1.84
i = 3^(1/1.84) = 1.82
j = 3^(1/1.82) = 1.83
k = 3^(1/1.83) = 1.82

1.82^1.82 ≈ 3

So after 11 iterations, we have 3 correct digits of the 2nd super-root of 3. After infinitely many iterations, we have infinitely many correct digits.


Thoughts? Has this been recognized before, and if so, where can I read about it? As far as I have read, square super-roots should not be able to be calculated with elementary functions.
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December 1st, 2017, 03:54 AM   #2
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Such a function can not be calculated using a finite number of elementary functions. But any function can be calculated as a limit of an infinite sequence of elementary functions.
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