Great EXPectations
July 13, 2010
21 comments
Given that x^(x^(x^(x^(x^(x^(x^(x^(x…)))…) = 2, solve for x.
Communicated by J. Goldfoot.
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Warning: Solutions May Be Discussed in the Comments
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Written by Korbin on June 21, 2011. Reply
what do the dots mean? can u email me and write out the whole question without simplifying it or shortening it
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Written by Steven Miller on June 21, 2011. Reply
It means the process continues. Take the limit of x, x^x, x^(x^x), x^(x^(x^x)), ….
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Written by Some guy on November 20, 2011. Reply
X = ∞√2
^That does not say infinity times root 2. I just don’t know how to type unicode for infinity root 2 or however it’s supposed to be said
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Written by Steven Miller on November 21, 2011. Reply
There’s a very simple solution. Email me at [email protected] if you want the soln or a hint.
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Written by Steven Miller on December 1, 2011. Reply
It’s not 2/3…. email me at [email protected] if you want a hint
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Written by Anon on December 31, 2011. Reply
let a^x = x^(x^(x^(x^…))…)
so a^x = 2
Take logs of both sides.
ln(a^x) = ln(2)
x*ln(a) = ln(2)
x = ln(2)/ln(a)
But since a is x to the power of x an infinite number of times, a = 2. So ln(a) = ln(2).Therefore x = 1.
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Written by Steven Miller on December 31, 2011. Reply
Anon: you’re close but not quite there — email me at [email protected] if you want to discuss more.
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Written by Terry on July 23, 2012. Reply
As I’m doing this on my phone please forgive typos and lack of proper notation. As x tends toward infinity then the answer tends toward 1. So the limit of x^x^x = 2 as x tends to infinity is x tending toward 1. Isn’t it??
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Written by Steven Miller on July 24, 2012. Reply
I don’t understand: if x tends to infinity how can x tend to 1? email me at sjm1 AT williams.edu
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Written by Steven Miller on October 1, 2012. Reply
anonymous: please include name/email: correct (sjm1 AT williams.edu)
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