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siriusmart

[2024/05/03] Infinite exponential (lemmy.world)

submitted 10 months ago by siriusmart to c/dailymaths

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Find x for x^x^x^x^x^... = 2

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[–] [email protected] 15 points 10 months ago* (last edited 10 months ago) (1 children)

First off, why did this post get downvoted?

Answer alert

The answer is sqrt(2), because sqrt(2)^2^ = 2, and you build the infinite exponential sequence by taking x_n = x^x_(n-1)^.

[–] siriusmart 11 points 10 months ago

yup there u go, btw can u please spoiler it so its harder for people to accidentally see it

[–] [email protected] 9 points 10 months ago (1 children)

Now try x^x^x^… = 4. And then tell me why it’s not a logical crisis that you get exactly the same answer when you solve x^x^x^… = 2 instead.

[–] [email protected] 7 points 10 months ago* (last edited 10 months ago) (1 children)

The reason is that x^x^... doesn't converge to anything greater than e.

Here's what happens with sqrt(2) as x with a 12 large power tower (WolframAlpha doesn't like power towers with roots):

https://www.wolframalpha.com/input?i2d=true&i=Nest+%5C%2891%29Power%5Bsqrt%5C%2840%292%5C%2841%29%2C%23%5D%26%5C%2844%29+1%5C%2844%29+12%5C%2893%29

Here's what happens with e^(1/e)^, the max value for x with a 500,000 large power tower:

https://www.wolframalpha.com/input?i2d=true&i=Nest+%5C%2891%29Power%5B%5C%2840%291.44466786%5C%2841%29%2C%23%5D%26%5C%2844%29+1%5C%2844%29+500000%5C%2893%29

(Note: I was unable to enter that value so I approximated it. The exact result should be e)

Here's what happens with the value above +0.00001 and a mere 1023 large power tower:

https://www.wolframalpha.com/input?i2d=true&i=Nest+%5C%2891%29Power%5B%5C%2840%291.44467787%5C%2841%29%2C%23%5D%26%5C%2844%29+1%5C%2844%29+1023%5C%2893%29

You can read the reasons here, I couldn't follow for very much but maybe you can get something out of it:

https://mathworld.wolfram.com/PowerTower.html

[–] [email protected] 1 points 10 months ago

What is weird is that x^x^x^x^.... does have multiple results as for e.g. x=√2 it can yield 2, 4, and possibly more. Maybe it is due to the concept of infinity that we cannot really grasp.

[–] [email protected] 8 points 10 months ago* (last edited 10 months ago)

spoiler

Allow y = x^x^x^...
Notice the exponent tower contains itself, such that:
y = x^y
Raise both sides to 1/y power:
y^(1/y) = x^(y/y)
y^(1/y) = x

For y = 2,
2^(1/2) = x
Aka:
x = √2