STD 12 - 7.2 definite integral — JEE STD 11 Science — Question

$ = \mathop {\lim }\limits_{n \to \infty } \,{\left( {\frac{1}{n}\,.\,\frac{2}{n}\,.\,\frac{3}{n}\,.\,\frac{4}{n}\,..........\frac{n}{n}} \right)^{1/n}}$

$\therefore \,\,\,\log \,\,P = \frac{1}{n}\,\mathop {\lim }\limits_{n \to \infty } \,\left( {\log \frac{1}{n} + \log \frac{2}{n} + ...... + \log \frac{n}{n}} \right)$

$\log \,\,P = \mathop {\lim }\limits_{n \to \infty } \,\sum\limits_{r = 1}^n {} \frac{1}{n}\log \frac{r}{n}$

$\log \,\,P = \int_0^1 {} \log x\,dx = (x\,\log x - x)_0^1 = ( - 1)$

==> $P = \frac{1}{e}$ .