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September 6th, 2015, 04:58 AM  #1 
Senior Member Joined: Sep 2015 From: 4th Dimension Posts: 146 Thanks: 13 Math Focus: Everything (a little bit)  let's see who is the fastest
Find the value of $\displaystyle 2!!!!!!!!!!!!!!!!!!............................... ...........................!!!!!!!!!!!! \text{(Upto infinity)}$ Last edited by skipjack; September 6th, 2015 at 06:26 AM. 
September 6th, 2015, 04:59 AM  #2 
Senior Member Joined: Sep 2015 From: 4th Dimension Posts: 146 Thanks: 13 Math Focus: Everything (a little bit) 
Is it me??? 
September 7th, 2015, 10:47 AM  #3 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
I have no idea what you are talking about. How do measure the time taken to look at that and say "that number does not exist"? (No, your signature is not interesting.) 
September 7th, 2015, 09:53 PM  #4 
Senior Member Joined: Sep 2015 From: 4th Dimension Posts: 146 Thanks: 13 Math Focus: Everything (a little bit)  
September 7th, 2015, 11:52 PM  #5 
Math Team Joined: Nov 2014 From: Australia Posts: 689 Thanks: 244 
Well, $2! = 2$, so if we take factorials a finite number of times, we get $$2!!!!...! = 2.$$ However, I have no idea what would happen with an infinite number of factorials (or even if the concept is well defined). 
September 8th, 2015, 10:04 AM  #6 
Senior Member Joined: Sep 2015 From: 4th Dimension Posts: 146 Thanks: 13 Math Focus: Everything (a little bit) 
We can prove that it is true for every $n+1$ and hence it is valid upto $\infinity$

September 9th, 2015, 01:27 AM  #7 
Math Team Joined: Nov 2014 From: Australia Posts: 689 Thanks: 244 
No. You can conclude that it is true for any positive finite number of factorials. The problem I have is that it is not clearly defined for an infinite number of factorials. One way to formally define it would be: Find $\displaystyle\lim_{n\to\infty}a_n,$ where $$a_1 = 2,\qquad\qquad a_{n + 1} = a_n!$$ 

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