August 23rd, 2019, 02:12 AM  #1 
Senior Member Joined: Dec 2015 From: somewhere Posts: 642 Thanks: 91  Prove inequality
Prove: $\displaystyle e^{n^2} \left(1^1 \cdot 2^2 \cdot 3^3 \cdot ...\cdot n^n \right)\leq n^{n^2 + n + 6}$.
Last edited by skipjack; August 23rd, 2019 at 06:16 AM. 
August 23rd, 2019, 09:35 AM  #2 
Senior Member Joined: Dec 2015 From: somewhere Posts: 642 Thanks: 91  
August 23rd, 2019, 09:49 AM  #3 
Senior Member Joined: Dec 2015 From: somewhere Posts: 642 Thanks: 91 
$\displaystyle e^{n^2 }(\underbrace{n^n •...•n^n }_{n} )<n^{n^2 +n+ 6}$. $\displaystyle (en)^{n^2 } < n^{n^2 } n^{n} n^{6} $. $\displaystyle e^{n^2 } <n^{n+6}\; $ , how to continue from here ? Seems like the inequality is not true ! Last edited by idontknow; August 23rd, 2019 at 09:58 AM. 
August 23rd, 2019, 09:49 AM  #4 
Math Team Joined: May 2013 From: The Astral plane Posts: 2,272 Thanks: 942 Math Focus: Wibbly wobbly timeywimey stuff.  
August 23rd, 2019, 01:18 PM  #5 
Global Moderator Joined: Dec 2006 Posts: 20,972 Thanks: 2222 
I've closed this thread as the original poster realizes he didn't get it right.


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