First slide

Introduction to definite integration

Question

${Lt}_{n \to \infty} {(\frac{n!}{n})}^{1 / n} is equal to$

Moderate

A

e
B

$\frac{1}{e}$
C

$e - 1$
D

$e - 5$

Solution

$\begin{array}{l} y = \lim_{n \to \infty} {(\frac{n!}{n})}^{1 / n} = \lim_{n \to \infty} {(\frac{n!}{n^{n}})}^{\frac{1}{n}} \\ \log y = \lim_{n \to \infty} \frac{1}{n} \cdot \log [\frac{1}{n} \cdot \frac{2}{n} \cdot \frac{3}{n} \dots \cdot \frac{n}{n}] \\ = \lim_{n \to \infty} \frac{1}{n} [\log \frac{1}{n} + \log \frac{2}{n} + \log \frac{3}{n} + \dots . . + \log \frac{n}{n}] \\ = \underset{n \to \infty}{L t} \sum_{r = 1}^{\infty} \frac{1}{n} \log (\frac{r}{n}) = \int_{0}^{1} \log x \cdot d x \\ = (x \log x - x)_{0}^{1} \\ \log_{e}^{y} = - 1 \\ \Rightarrow y = e^{- 1} \Rightarrow y = \frac{1}{e} \end{array}$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App