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June 24th, 2010, 06:28 AM   #1
 
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Does this equal e?

It is the sum of the inverse of a modification of the factorial function,
where exponentiation is used instead of multiplication.



1. Is the above statement true?
2. Any tips on how I should prove it?

PS. Sorry if I've chosen the wrong category, I'm not sure where this belongs.
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June 24th, 2010, 06:36 AM   #2
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Re: Does this equal e?

e is larger. The factorial grows much more slowly than the tetration-like function you use in the denominator.

The value of your expression is almost exactly 6160393/2359296 = 2.611114925808376736111111.... The approximation is correct to over a hundred thousand decimal places.
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June 24th, 2010, 06:43 AM   #3
 
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Re: Does this equal e?

Oh, I agree that was obvious.
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June 24th, 2010, 06:57 AM   #4
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Re: Does this equal e?

I don't know whether it helps you, but I assume, you know:


Your function is basicly
.

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June 24th, 2010, 07:21 AM   #5
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Re: Does this equal e?

Quote:
Originally Posted by Hoempa
Your function is basicly
.
That wasn't what I figured at all. That's double exponential (the k-th term is about ) where what I was looking at was tetrational (the k-th term was about with k ks.)
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June 24th, 2010, 07:30 AM   #6
 
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Re: Does this equal e?

We have Sigma for addition, Pi for multiplication, but we don't have anything for exponentiation.
Is there a notation for integer-repetition where one could specify the operation itself?
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June 24th, 2010, 07:35 AM   #7
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Re: Does this equal e?

Can't you use like I did, or is it false? I tried to describe brangelito's function, and I do agree e is larger.
Brangelito, do you mean this

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June 24th, 2010, 07:39 AM   #8
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Re: Does this equal e?

Quote:
Originally Posted by brangelito
We have Sigma for addition, Pi for multiplication, but we don't have anything for exponentiation.
Is there a notation for integer-repetition where one could specify the operation itself?
No, not really. The normal way of representing this would be with recursion: f(1) = 1, f(n) = n^f(n-1).

Quote:
Originally Posted by Hoempa
Brangelito, do you mean this
That's what I use to represent tetration when all the numbers are the same. In this case they aren't, though...
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June 24th, 2010, 07:41 AM   #9
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Re: Does this equal e?

Quote:
Originally Posted by Hoempa
Can't you use like I did, or is it false?
That's false. a^(b^(c^d)) which is in general not equal to ((a^b)^c)^d since exponentiation is not associative. 3^(3^3) = 7625597484987, while (3^3)^3 = 729.
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June 24th, 2010, 07:50 AM   #10
 
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Re: Does this equal e?

I actually meant the tetration way, or at least that's what I calculated.
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