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I have been studying exponential factorials and have been looking for the equivalent tetration. For example:
5^4^3^2^1= 5.9 e16
10^9^8^7^6^5^4^3^2^1 = 10 e363879
I believe as n goes to infinity the exponential factorial can be written as a tetration:
n^(n-1)^(n-2)...^2^1 =
(n/alpha)^(n/alpha)^(n/alpha)^(n/alpha)...repeated n times
where alpha is Feigenbaum constant 2.5029...
I have been testing this on this site:
http://www.ttmath.org/online_calculator
It does seem very very close for up to 25^24^23^...^2^1 can be written as an equivalent tetration. I would anyones input to see if they can help me determine it can be written as a tetration more rigorously. Thanks very much
Ryan Gerard
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(11/12/2009, 10:42 PM)rsgerard Wrote: I have been studying exponential factorials and have been looking for the equivalent tetration. For example:
5^4^3^2^1= 5.9 e16
10^9^8^7^6^5^4^3^2^1 = 10 e363879
Well what you are saying is not really the exponential factorial.
What you wrote above is the same as \( x^{(x-1)!} \)
The value of \( 5^{4^{3^{2^1}}} \) is acutaly more like 6.206e+183230
As for
(11/12/2009, 10:42 PM)rsgerard Wrote: n^(n-1)^(n-2)...^2^1 =
(n/alpha)^(n/alpha)^(n/alpha)^(n/alpha)...repeated n times
I would have to look into this more to see how close to \( x^{(x-1)!} \) it is
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11/12/2009, 11:19 PM
(This post was last modified: 11/12/2009, 11:21 PM by nuninho1980.)
aah! it's normal. sorry. I removed.
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Honestly, I have never heard of the Feigenbaum constant before. Is 2.5029 the value for exponential functions? or all functions?
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11/12/2009, 11:51 PM
(This post was last modified: 11/12/2009, 11:53 PM by andydude.)
(11/12/2009, 11:19 PM)dantheman163 Wrote: Well what you are saying is not really the exponential factorial.
Correct.
5^4^3^2^1 = 5^(4^(3^(2^1))) = \( 5^{4^{3^{2^1}}} \ne 5^{(4\cdot 3 \cdot 2 \cdot 1)} \) = (((5^4)^3)^2)^1.
I think the problem here is that ttmath.org evaluates (a^b^c) incorrectly. It should use right-associative (^), but it seems to use left-associative (^), which is wrong.
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(11/12/2009, 11:51 PM)andydude Wrote: (11/12/2009, 11:19 PM)dantheman163 Wrote: Well what you are saying is not really the exponential factorial.
Correct.
5^4^3^2^1 = 5^(4^(3^(2^1))) = \( 5^{4^{3^{2^1}}} \ne 5^{(4\cdot 3 \cdot 2 \cdot 1)} \) = (((5^4)^3)^2)^1.
I think the problem here is that ttmath.org evaluates (a^b^c) incorrectly. It should use right-associative (^), but it seems to use left-associative (^), which is wrong.
Thanks for pointing out that this site is using left-association to evaluate. I guess I'm still curious to see if this constant really applies to these functions. I'll do a little more research myself. Thanks so much everyone.
