JEE Mains · Maths · STD 12 - 7.2 definite integral
\(\lim _{n \rightarrow \infty} \frac{1}{2^{n}}\left(\frac{1}{\sqrt{1-\frac{1}{2^{a}}}}+\frac{1}{\sqrt{1-\frac{2}{2^{n}}}}+\frac{1}{\sqrt{1-\frac{3}{2^{a}}}}+\ldots \ldots+\frac{1}{\sqrt{1-\frac{2^{a}-1}{2^{n}}}}\right)\) is equal to
- A \(\frac{1}{2}\)
- B \(1\)
- C \(2\)
- D \(-2\)
Answer & Solution
Correct Answer
(C) \(2\)
Step-by-step Solution
Detailed explanation
\(I=\lim _{n \rightarrow \infty} \frac{1}{2^{n}}\left(\frac{1}{\sqrt{1-\frac{1}{2^{n}}}}+\frac{1}{\sqrt{1-\frac{2}{2^{n}}}}+\frac{1}{\sqrt{1-\frac{3}{2^{n}}}}+\ldots .+\frac{1}{\sqrt{1-\frac{2^{n}-1}{2^{n}}}}\right)\) Let \(2^{n}=t\) and if \(n \rightarrow \infty\) then…
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
- Let a line \(L\) pass through the point of intersection of the lines \(b x+10 y-8=0\) and \(2 x-3 y=0\), \(b \in R -\left\{\frac{4}{3}\right\}\). If the line \(L\) also passes through the point \((1,1)\) and touches the circle \(17\left( x ^{2}+ y ^{2}\right)=16\), then the eccentricity of the ellipse \(\frac{x^{2}}{5}+\frac{y^{2}}{b^{2}}=1\) is.JEE Mains 2022 Hard
- The shortest distance between the \(z-\) axis and the line \(x + y + 2z - 3\, = 0 \,= 2x + 3y + 4z - 4\), isJEE Mains 2015 Hard
- The sum of the roots of the equation \(x+1-2 \log _{2}\left(3+2^{x}\right)+2 \log _{4}\left(10-2^{-x}\right)=0\), is :JEE Mains 2021 Hard
- Let \(\vec a\, = \,\hat i\, + \,\hat j\, + \,\sqrt 2 \hat k,\,\,\vec b\, = \,{b_1}\hat i\, + \,{b_2}\hat j\, + \sqrt 2 \hat k\) and \(\vec c\, = \,5\hat i\, + \,\hat j + \sqrt 2 \hat k\) be three vectors such that the projection vector of \(\vec b\) on \(\vec a\) is \(\vec a\). If \(\vec a\, + \vec b\) is perpendicular to \(\vec c\) , then \(\left| {\vec b} \right|\) is equal toJEE Mains 2019 Hard
- Let \([\cdot]\) denote the greatest integer function. Then the value of \(\displaystyle\int_0^3 \left(\dfrac{e^x + e^{-x}}{[x]!}\right) dx\) is :JEE Mains 2026 Medium
- If \({I_1} = \int\limits_0^1 {{e^{ - x}}} {\cos ^2}\,x\,dx\,;\,{I_2} = \int\limits_0^1 {{e^{ - {x^2}}}} {\cos ^2}\,x\,dx\) and \(\,{I_3} = \int\limits_0^1 {{e^{ - {x^3}}}} dx\) ; thenJEE Mains 2018 Hard
More PYQs from JEE Mains
- For an initial screening of an admission test, a candidate is given fifty problems to solve. If the probability that the candidate can solve any problem is \(\frac{4}{5}\) , then the probability that he is unable to solve less than two problems isJEE Mains 2019 Hard
- Let the foci and length of the latus rectum of an ellipse \(\frac{\mathrm{x}^2}{\mathrm{a}^2}+\frac{\mathrm{y}^2}{\mathrm{~b}^2}=1, \mathrm{a}>\mathrm{b}\) be \(( \pm 5,0)\) and \(\sqrt{50}\), respectively. Then, the square of the eccentricity of the hyperbola \(\frac{\mathrm{x}^2}{\mathrm{~b}^2}-\frac{\mathrm{y}^2}{\mathrm{a}^2 \mathrm{~b}^2}=1\) equalsJEE Mains 2024 Hard
- If \(\mathrm{R}\) is the smallest equivalence relation on the set \(\{1,2,3,4\}\) such that \(\{(1,2),(1,3)\} \subset R\), then the number of elements in \(\mathrm{R}\) isJEE Mains 2024 Medium
- Let \(y=y(x)\) be the solution of the differential equation \(\left(x-x^{3}\right) d y=\left(y+y x^{2}-3 x^{4}\right) d x, x>2\). If \(y(3)=3\), then \(y(4)\) is equal to :JEE Mains 2021 Hard
- If \(y=y(x)\) is the solution of the differential equation,
\(\sqrt{4-x^2} \frac{\mathrm{~d} y}{\mathrm{~d} x}=\left(\left(\sin ^{-1}\left(\frac{x}{2}\right)\right)^2-y\right) \sin ^{-1}\left(\frac{x}{2}\right),-2 \leq x \leq 2, y(2)=\frac{\pi^2-8}{4}\), then \(y^2(0)\) is equal toJEE Mains 2025 Medium - Group A consists of 7 boys and 3 girls, while group B consists of 6 boys and 5 girls. The number of ways, 4 boys and 4 girls can be invited for a picnic if 5 of them must be from group \(A\) and the remaining 3 from group \(B\), is equal to :JEE Mains 2025 Easy