MHB Comparing Factorials and Exponentials: Which is Greater?

[anemone]

Determine which of the following is greater?

$2015!$ or $1008^{2015}$?

 

[kaliprasad]

Determine which of the following is greater?

$2015!$ or $1008^{2015}$?

Spoiler

we have
$1008^2 \gt (1008^2-n^2) \gt(1008-n)(1008+n)$
by taking n from 1007 to 1 we get
$1008^{2014}$ on LHS and $ 1 * 2 * \cdots 1007 * 2015 * \cdots * 1009 = \dfrac{2015!}{1008}$ on RHS
hence
$1008^{2014}\gt \dfrac{2015!}{1008} $
or $1008^{2015}\gt 2015! $
hence $1008^{2015}$ is greater

 

[anemone]

Spoiler

we have
$1008^2 \gt (1008^2-n^2) \gt(1008-n)(1008+n)$
by taking n from 1007 to 1 we get
$1008^{2014}$ on LHS and $ 1 * 2 * \cdots 1007 * 2015 * \cdots * 1009 = \dfrac{2015!}{1008}$ on RHS
hence
$1008^{2014}\gt \dfrac{2015!}{1008} $
or $1008^{2015}\gt 2015! $
hence $1008^{2015}$ is greater

Very great job, kaliprasad!