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 October 18th, 2017, 09:19 AM #1 Senior Member   Joined: Dec 2015 From: Earth Posts: 256 Thanks: 28 n! inequality How to prove if $\displaystyle \; {(\frac{n}{3})}^{n}\leq n!$
 October 18th, 2017, 09:32 AM #2 Member     Joined: Aug 2011 From: Nouakchott, Mauritania Posts: 85 Thanks: 14 Math Focus: Algebra, Cryptography Salam ! Did you try to show this inequality by induction ? You may use the Binomial Expansion.
 October 18th, 2017, 09:56 AM #3 Senior Member   Joined: Sep 2016 From: USA Posts: 502 Thanks: 280 Math Focus: Dynamical systems, analytic function theory, numerics It isn't a true statement so it would be hard to prove.
October 18th, 2017, 10:29 AM   #4
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Quote:
 Originally Posted by SDK It isn't a true statement so it would be hard to prove.
In my book it says true, but I want to see how to get there?
And I forgot to write $\displaystyle n \in N$.

Last edited by skipjack; October 18th, 2017 at 12:14 PM.

 October 18th, 2017, 03:00 PM #5 Global Moderator   Joined: May 2007 Posts: 6,627 Thanks: 622 An approximate proof would start using Stirling's formula: $\displaystyle n!\simeq \sqrt {2\pi n}(\frac{n}{e})^n>(\frac{n}{3})^n$ Checking on the derivation of Stirling's formula. $\displaystyle \sqrt {2\pi n}(\frac{n}{e})^n$ is a lower bound for n!. Thanks from Ould Youbba and idontknow Last edited by mathman; October 19th, 2017 at 02:02 PM.

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