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\[\lim_{n \rightarrow \infty}\frac{ n! }{ n^n }\]

If I expand the factorial the first n term cancels but I'm really sure where to go from there.

|dw:1441847154717:dw|

Not sure how to keep going after that...

show that \(n!

Induction. Ohh boy.

I would begin by changing \[\frac{n!}{n^n}\] into \[n^{-n}n!\] then logarithms

\[\frac{n!}{n^n}=\frac{n(n-1)\cdots 32}{nn\cdots nn}\frac{1}{n}\le1\cdot \frac{1}{n}=\frac{1}{n}\]

Where did you get a 32 from?

...3*2*1

Ohh.

Nope still not clicking.

Wait I got it. it was just induction. Bah.

Aha! That's brilliant! I used induction but there seems to be 2 methods :) .