JEE Mains · Maths · STD 12 - 7.2 definite integral
\(\lim _{n \rightarrow \infty}\left[\frac{1}{n}+\frac{n}{(n+1)^{2}}+\frac{n}{(n+2)^{2}}+\ldots \ldots .+\frac{n}{(2 n-1)^{2}}\right]\) is equal to ...... .
- A \(\frac{1}{2}\)
- B \(1\)
- C \(\frac{1}{3}\)
- D \(\frac{1}{4}\)
Answer & Solution
Correct Answer
(A) \(\frac{1}{2}\)
Step-by-step Solution
Detailed explanation
\(\lim _{n \rightarrow \infty}\left[\frac{1}{n}+\frac{n}{(n+1)^{2}}+\frac{n}{(n+2)^{2}}+\ldots+\frac{n}{(2 n-1)^{2}}\right]\) \(=\lim _{n \rightarrow \infty} \sum_{r=0}^{n-1} \frac{n}{(n+r)^{2}}=\lim _{n \rightarrow \infty} \sum_{r=0}^{n-1} \frac{n}{n^{2}+2 n r+r^{2}}\)…
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
- A \(10\, inches\) long pencil \(\mathrm{AB}\) with mid point \(\mathrm{C}\) and a small eraser \(\mathrm{P}\) are placed on the horizontal top of a table such that \(\mathrm{PC}=\sqrt{5}\) inches and \(\angle \mathrm{PCB}=\tan ^{-1}(2)\). The acute angle through which the pencil must be rotated about \(\mathrm{C}\) so that the perpendicular distance between eraser and pencil becomes exactly \(1\, inch\) is:
JEE Mains 2021 Hard - The number of solutions of \(\sin ^{7} x+\cos ^{7}=1, x \in[0,4 \pi]\) is equal to :JEE Mains 2021 Hard
- If the system of equations \(\alpha x+y+z=5, x+2 y+\) \(3 z=4, x+3 y+5 z=\beta\) has infinitely many solutions, then the ordered pair \((\alpha, \beta)\) is equal to:JEE Mains 2022 Medium
- An ellipse passes through the foci of the hyperbola, \(9x^2 - 4y^2 = 36\) and its major and minor axes lie along the transverse and conjugate axes of the hyperbola respectively. If the product of eccentricities of the two conics is \(\frac {1}{2}\), then which of the following points does not lie on the ellipse?JEE Mains 2015 Hard
- Let \(\alpha\) and \(\beta\) be the roots of \(x^2+\sqrt{3 x}-16=0\), and \(\gamma\) and \(\delta\) be the roots of \(x^2+3 x-1=0\). If \(P_n=\alpha^n+\beta^n\) and \(Q_n=\gamma^n+\delta^n\), then \(\frac{\mathrm{P}_{25}+\sqrt{3 \mathrm{P}_{24}}}{2 \mathrm{P}_{23}}+\frac{\mathrm{Q}_{25}-\mathrm{Q}_{23}}{\mathrm{Q}_{24}}\) is equal toJEE Mains 2025 Medium
- Locus of the image of point \( (2,3)\) in the line \(\left( {2x - 3y + 4} \right) + k\left( {x - 2y + 3} \right) = 0,k \in R\) is a:JEE Mains 2015 Hard
More PYQs from JEE Mains
- If \(\lambda \in R\) is such that the sum of the cubes of the roots of the equation, \(x^2 +(2 - \lambda ) x+ (10 - \lambda ) = 0\) is minimum, then the magnitude of the difference of the roots of this equation isJEE Mains 2018 Hard
- If \(f(x)=\int_{0}^{x}(5+|1-t|) d t, \quad x>2\) \(\quad \quad \quad \quad \quad 5 x+1,\quad \quad \quad \quad \quad x \leq 2\), thenJEE Mains 2021 Hard
- \(\mathop {\lim }\limits_{x \to 0} \frac{{{{\left( {1 - \cos \,2x} \right)}^2}}}{{2x\,\tan \,x - x\,\tan \,2x}}\) isJEE Mains 2016 Hard
- Let the image of the point \(P(1, 6, a)\) in the line \(L: \dfrac{x}{1} = \dfrac{y-1}{2} = \dfrac{z-a+1}{b}\), \(b > 0\), be \(\left(\dfrac{a}{3}, 0, a+c\right)\). If \(S(\alpha, \beta, \gamma)\), \(\alpha > 0\), is the point on \(L\) such that the distance of \(S\) from the foot of perpendicular from the point \(P\) on \(L\) is \(2\sqrt{14}\), then \(\alpha + \beta + \gamma\) is equal to:JEE Mains 2026 Hard
- The term independent of \(x\) in the expression of \(\left(1-x^{2}+3 x^{3}\right)\left(\frac{5}{2} x^{3}-\frac{1}{5 x^{2}}\right)^{11}, x \neq 0\) isJEE Mains 2022 Hard
- Let \(\alpha\) and \(\beta\) be two real numbers such that \(\alpha+\beta=1\) and \(\alpha \beta=-1 .\) Let \(p _{ n }=(\alpha)^{ n }+(\beta)^{ n },p _{ n -1}=11\) and \(p _{ n +1}=29\) for some integer \(n \geq 1 .\) Then, the value of \(p _{ n }^{2}\) is .... .JEE Mains 2021 Hard