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

The time period of a particle in simple harmonic motion is $8 \ s$. At $t=0$,it is at the mean position. The ratio of the distances travelled by it in the first and second seconds is

Three rods made of the same material and having the same cross-section are joined as shown in the figure. Each rod is of the same length. The left and right ends are kept at $0^{\circ}C$ and $90^{\circ}C$ respectively. The temperature of the junction of the three rods will be ...... $^{\circ}C$.

An element $P$ has atomic number $56$. What will be the formula of its halide?

If $A, B$ and $C$ are mutually exclusive and exhaustive events of a random experiment such that $P(B) = \frac{3}{2} P(A)$ and $P(C) = \frac{1}{2} P(B)$,then $P(A \cup C)$ equals to

In the circuit shown,the heat produced in $5 \Omega$ resistance due to current $I_2$ is $50 \text{ J/s}$. Then,the heat generated per second in $2 \Omega$ resistance is (in $\text{ J/s}$)

Vedclass Test Series

Exam Paper Generator

Online Exam Module