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Prove that $\frac{n^n}{3^n} < n! < \frac{n^n}{2^n} $ for each $ n \geq 6 $

I want to prove $\dfrac{n^n}{3^n} \lt n! \lt \dfrac{n^n}{2^n}$ for all $n$ $\geq$ 6:

So we do the base case with $n$ = $6$ and it is really exciting.

Now for the induction hypothesis we assume the formula to be true. I started out by doing simple manipulations with $n+1$, however I get nowhere... (well I actually got close but my iff was actually just an if then...)

  1. How can I get this off of the ground?
  2. What pieces of information can I use?
  3. Thanks!
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    I swear i saw this within the past few weeks, here on MSE. I thought it was a nice introduction to Stirling's.2012-09-20

0 Answers 0