Limit as n approaches infinity of (log(n!) / nlog(n))

Question

I'm stumped with a limit. I know the answer (because I looked on wolframalpha) but I really want to know how to reach the correct answer. If you have any hints or tips on getting there I'd be most appreciative!

score 4 · Accepted Answer · answered Jan 19 '14 at 21:20

4

Hint:

$$ \ln(n!) = \sum_{k=1}^{n} \ln k \sim \int_{1}^{n} \ln x dx =n\ln n-n+1 .$$

answered Jan 19 '14 at 21:20

Mhenni Benghorbal

47,431

Limit as n approaches infinity of (log(n!) / nlog(n))

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