Definite integration...

Level 1

Find the value of $\displaystyle \lim_{n \rightarrow \infty } \left(\dfrac{n!}{n^{n}}\right)^{ \frac{1}{n}}$

where $n! = n\ \times (n-1) \times (n-2).... \times 2 \times 1$ .

e

\infty

e^{2}

\frac{1}{e}

1 solution

Brian Charlesworth
Oct 22, 2017

The natural log of the given limit is $\displaystyle\lim_{n \to \infty} \dfrac{1}{n}\times (\ln(n!) - \ln(n^{n})) = \lim_{n \to \infty} \dfrac{1}{n} \times (ln(n!) - n\ln(n))$ .

Now by Stirling's approximation $ln(n!) \approx n\ln(n) - n$ for large $n$ , so our log limit then becomes $\displaystyle\lim_{n \to \infty} \dfrac{1}{n} \times (n\ln(n) - n - n\ln(n)) = -1$ .

The original limit is then $e^{-1} = \boxed{\dfrac{1}{e}}$ .

Definite integration...

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