AP EAMCET · Maths · Sequences and Series
The value of \(\lim _{\substack{n \rightarrow \infty}} \frac{1}{n^3} \sum_{k=1}^n\left(k^2 x\right)\) is
- A \(x\)
- B \(\frac{x}{2}\)
- C \(\frac{x}{3}\)
- D \(\frac{x}{4}\)
Answer & Solution
Correct Answer
(C) \(\frac{x}{3}\)
Step-by-step Solution
Detailed explanation
\(\begin{aligned} \lim _{n \rightarrow \infty} \frac{1}{n^3} \sum_{k=1}^n\left(k^2\right. & x) \\ & =x \lim _{n \rightarrow \infty} \frac{1}{n} \sum_{k=1}^n\left(\frac{k}{n}\right)^2 \\ & =x \int_0^1 x^2 d x=x\left[\frac{x^3}{3}\right]_0^1 \\ & =\frac{x}{3}\end{aligned}\)
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- The equation of the pair of lines joining the origin to the points of intersection of \(x^2+y^2=9\) and \(x+y=3\), isAP EAMCET 2016 Medium
- Which of the following set of data has least standard deviation?AP EAMCET 2021 Easy
- If , then is equal toAP EAMCET 2021 Easy
- \(\lim _{x \rightarrow \frac{\pi}{2}} \frac{1+\cos 2 x}{\cot 3 x\left(3^{\sin 2 x}-1\right)}=\)AP EAMCET 2018 Medium
- In a triangle if the mid points of sides are , respectively, thenAP EAMCET 2019 Easy
- If \(B\) is the inverse of \(a\) third order matrix \(A\) and \(\operatorname{det} B=k\), then \((\operatorname{adj}(\operatorname{adj} A))^{-1}=\)AP EAMCET 2025 Medium
More PYQs from AP EAMCET
- The sum of two lower triangular matrices is alwaysAP EAMCET 2021 Easy
- An object of mass \(2 \mathrm{~kg}\) is attached to a spring with spring constant \(8 \mathrm{Nm}^{-1}\). If the object is executing simple harmonic motion then the number of cycles it completes in \(66 \mathrm{~s}\) isAP EAMCET 2022 Easy
- \[
\int_0^{\pi^2 / 4}(2 \sin \sqrt{x}+\sqrt{x} \cos \sqrt{x}) d x=
\]AP EAMCET 2022 Medium - IUPAC names of the following compounds A and B are
AP EAMCET 2024 Medium - If \(\left[\begin{array}{rrr}1 & -1 & x \\ 1 & x & 1 \\ x & -1 & 1\end{array}\right]\) has no inverse, then the real value of \(x\) isAP EAMCET 2009 Easy
- Two slabs \(A\) and \(B\) of different materials but of the same thickness are joined end to end to form a composite slab. The thermal conductivities of \(A\) and \(B\) are \(k_1\) and \(k_2\) respectively. A steady temperature difference of \(12^{\circ} \mathrm{C}\) is maintained across the composite slab. If \(k_1=\frac{k_2}{2}\), the temperature difference across slab \(A\) isAP EAMCET 2011 Easy