lim _{n rightarrow infty} frac{1}{n} left{ se | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) આપેલ લક્ષ: $\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{r=1}^{n} \sec ^{2} \left( \frac{r \pi}{4 n} \right)$.નિશ્ચિત સંકલનની વ્યાખ્યા મુજબ,$\li

Vedclass · Accepted Answer

$\frac{4}{\pi}$

Vedclass · Answer

(C) આપેલ લક્ષ: $\lim _{n ightarrow \infty} \frac{1}{n} \sum_{r=1}^{n} \sec ^{2} \left( \frac{r \pi}{4 n} ight)$.નિશ્ચિત સંકલનની વ્યાખ્યા મુજબ,$\lim _{n ightarrow \infty} \frac{1}{n} \sum_{r=1}^{n} f \left( \frac{r}{n} ight) = \int_{0}^{1} f(x) dx$.અહીં,$f(x) = \sec ^{2} \left( \frac{\pi x}{4} ight)$.તેથી,સંકલન $\int_{0}^{1} \sec ^{2} \left( \frac{\pi x}{4} ight) dx$ થશે.$\sec ^{2} \left( \frac{\pi x}{4} ight)$ નું સંકલન કરતા,આપણને $\frac{4}{\pi} 	an \left( \frac{\pi x}{4} ight)$ મળે છે.$0$ થી $1$ ની મર્યાદામાં નિશ્ચિત સંકલનનું મૂલ્ય: $\left[ \frac{4}{\pi} 	an \left( \frac{\pi x}{4} ight) ight]_{0}^{1} = \frac{4}{\pi} \left( 	an \frac{\pi}{4} - 	an 0 ight)$.કારણ કે $	an \frac{\pi}{4} = 1$ અને $	an 0 = 0$,તેથી કિંમત $\frac{4}{\pi} (1 - 0) = \frac{4}{\pi}$ થાય છે.

$\lim _{n \rightarrow \infty} \frac{1}{n} \left\{ \sec ^{2} \frac{\pi}{4 n} + \sec ^{2} \frac{2 \pi}{4 n} + \ldots + \sec ^{2} \frac{n \pi}{4 n} \right\}$ ની કિંમત શોધો.

Similar Questions

$\lim _{n}$ ${\rightarrow \infty}\left[\left(1+\frac{1}{n^2}\right)\left(1+\frac{2^2}{n^2}\right) \ldots\left(1+\frac{n^2}{n^2}\right)\right]^{\frac{1}{n}}=$

$\mathop {\lim }\limits_{n \to \infty } \left( {\frac{{n + 1}}{{{n^2} + {1^2}}} + \frac{{n + 2}}{{{n^2} + {2^2}}} + \frac{{n + 3}}{{{n^2} + {3^2}}} + \dots + \frac{1}{n}} \right) = $

$\lim _{n \rightarrow \infty}\left[\frac{n}{(n+1) \sqrt{2n+1}}+\frac{n}{(n+2) \sqrt{2(2n+2)}}+\frac{n}{(n+3) \sqrt{3(2n+3)}}+\ldots n \text{ પદો}\right]=\int_0^1 f(x) d x$,તો $f(x)=$

$\int_0^3 (2+x^2) dx = $

$\mathop {\lim }\limits_{n \to \infty } \frac{{{1^p} + {2^p} + {3^p} + ..... + {n^p}}}{{{n^{p + 1}}}} = $

Vedclass Test Series

Exam Paper Generator

Online Exam Module