June 24th, 2010, 06:28 AM  #1 
Senior Member Joined: Apr 2010 Posts: 215 Thanks: 0  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. 
June 24th, 2010, 06:36 AM  #2 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 930 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Does this equal e?
e is larger. The factorial grows much more slowly than the tetrationlike 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. 
June 24th, 2010, 06:43 AM  #3 
Senior Member Joined: Apr 2010 Posts: 215 Thanks: 0  Re: Does this equal e?
Oh, I agree that was obvious.

June 24th, 2010, 06:57 AM  #4 
Math Team Joined: Apr 2010 Posts: 2,765 Thanks: 351  Re: Does this equal e?
I don't know whether it helps you, but I assume, you know: Your function is basicly . Hoempa 
June 24th, 2010, 07:21 AM  #5  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 930 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Does this equal e? Quote:
 
June 24th, 2010, 07:30 AM  #6 
Senior Member Joined: Apr 2010 Posts: 215 Thanks: 0  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 integerrepetition where one could specify the operation itself? 
June 24th, 2010, 07:39 AM  #8  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 930 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Does this equal e? Quote:
Quote:
 
June 24th, 2010, 07:41 AM  #9  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 930 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Does this equal e? Quote:
 
June 24th, 2010, 07:50 AM  #10 
Senior Member Joined: Apr 2010 Posts: 215 Thanks: 0  Re: Does this equal e?
I actually meant the tetration way, or at least that's what I calculated.


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