The value of lim_{n rightarrow infty} frac{1} | vedclass

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

Question

(C) We are given the limit: $\lim_{n \rightarrow \infty} \frac{1}{n^3} \sum_{k=1}^n (k^2 x)$.Since $x$ is independent of $k$,we can take it out of the

Vedclass · Accepted Answer

$\frac{x}{3}$

Vedclass · Answer

(C) We are given the limit: $\lim_{n \rightarrow \infty} \frac{1}{n^3} \sum_{k=1}^n (k^2 x)$.Since $x$ is independent of $k$,we can take it out of the summation: $x \lim_{n \rightarrow \infty} \frac{1}{n} \sum_{k=1}^n \left(\frac{k}{n}\right)^2$.Using the definition of a definite integral as the limit of a Riemann sum,$\lim_{n \rightarrow \infty} \frac{1}{n} \sum_{k=1}^n f\left(\frac{k}{n}\right) = \int_0^1 f(t) dt$.Here,$f(t) = t^2$,so the expression becomes: $x \int_0^1 t^2 dt$.Evaluating the integral: $x \left[ \frac{t^3}{3} \right]_0^1 = x \left( \frac{1}{3} - 0 \right) = \frac{x}{3}$.

The value of $\lim_{n \rightarrow \infty} \frac{1}{n^3} \sum_{k=1}^n (k^2 x)$ is

Similar Questions

Consider the situation shown in the figure. The wire $AB$ is sliding on the fixed rails with a constant velocity $V$. If the wire $AB$ is replaced by a semicircular wire of the same chord length,the magnitude of the induced current will

Which of the following reagents will convert propanoic acid to propan-$1$-ol?

$A$ convex lens of focal length $F$ produces a real image $n$ times the size of the object. The image distance is

The ratio of momenta of an electron and an $\alpha$-particle which are accelerated from rest by a potential difference of $100\, V$ is

$A$ body is rolling down an inclined plane. If the kinetic energy of rotation is $40\%$ of the translational kinetic energy,then the body is a

Vedclass Test Series

Exam Paper Generator

Online Exam Module