$Compute $\lim_{n \rightarrow \infty} \ln \frac{n!}{n^n}$$

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$Compute $\lim_{n \rightarrow \infty} \ln \frac{n!}{n^n}$$

I'd like to compute the limit

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

I know I should be using Reimann sums to do it, but I am not sure how. Thanks.

Calculus Logarithms Limits

User35

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Note that
\[\lim_{n \rightarrow \infty}\frac{n!}{n^n}=\lim_{n \rightarrow \infty}\frac{1}{n}\times \frac{2 \times 3 \times 4\times \dots \times n}{n \times n \times \dots \times n}\leq \lim_{n \rightarrow \infty}\frac{1}{n} =0.\]
So
\[\lim_{n \rightarrow \infty}\frac{n!}{n^n}=0,\]
and
\[\lim_{n \rightarrow \infty}\ln \frac{n!}{n^n}=- \infty.\]

Erdos

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$Compute $\lim_{n \rightarrow \infty} \ln \frac{n!}{n^n}$$

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