AP EAMCET · Maths · Definite Integration
\(\lim _{n \rightarrow \infty}\left(\frac{1}{1^2+n^2}+\frac{2}{2^2+n^2}+\frac{3}{3^2+n^2}+\ldots+\frac{n}{n^2+n^2}\right)=\)
- A 1
- B \(\frac{1}{2} \log 2\)
- C \(2 \log 2\)
- D 0
Answer & Solution
Correct Answer
(B) \(\frac{1}{2} \log 2\)
Step-by-step Solution
Detailed explanation
\(\lim _{n \rightarrow \infty}\sum_{k=1}^{n} \frac{k}{k^2+n^2} = \lim _{n \rightarrow \infty}\sum_{k=1}^{n} \frac{1}{n} \frac{\frac{k}{n}}{(\frac{k}{n})^2+1}\) \(= \int_{0}^{1} \frac{x}{x^2+1} dx\) \(= \frac{1}{2} \int_{0}^{1} \frac{2x}{x^2+1} dx\)…
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
- If the extreme value of the function \(f(x)=\frac{4}{\sin x}+\frac{1}{1-\sin x}\) in \(\left[0, \frac{\pi}{2}\right]\) is \(m\) and it exists at \(\mathrm{x}=\mathrm{k}\), then \(\cos \mathrm{k}=\)AP EAMCET 2025 Medium
- The pole of the straight line \(x+4 y=4\) with respect to the ellipse \(x^2+4 y^2=4\) isAP EAMCET 2002 Easy
- If the function \(f: R \rightarrow R\) is defined by \(f(x)=x|x|\), then .AP EAMCET 2022 Easy
- If \(\int e^x(1+x) \cdot \sec ^2\left(x e^x\right) d x\) \(=f(x)+\) constant, then \(f(x)\) is equal toAP EAMCET 2008 Medium
- Number of unit vectors of the form where isAP EAMCET 2020 Easy
- \(\frac{d}{d x}\left(e^{\log _e \sqrt{1+\tan ^2 x}}\right)=\)AP EAMCET 2020 Medium
More PYQs from AP EAMCET
- If the pair of straight lines \(x y-x-y+1=0\) and the line \(a x+2 y-3 a=0\) are concurrent, then \(a\) is equal toAP EAMCET 2002 Medium
- \[
\begin{aligned}
& \text { If } \int \cos x \cdot \cos 2 x \cdot \cos 5 x d x \\
& =A \sin 2 x+B \sin 4 x+C \sin 6 x+D \sin 8 x+k
\end{aligned}
\]
(where \(k\) is the arbitrary constant of integration), then \(\frac{1}{B}+\frac{1}{C}=\)AP EAMCET 2019 Hard - In the given circuit values of \(I_1, I_2, I_3\) are respectively
AP EAMCET 2022 Hard - An extreme value of \(f(x)=\frac{4}{\sin x}+\frac{1}{1-\sin x}\) in \((0, \pi / 2)\) isAP EAMCET 2023 Medium
- Compound ' X ' is prepared commercially by the electrolysis of brine solution. Which of the following is not the use of ' X '?AP EAMCET 2025 Easy
- The total number of structural products formed in the following reaction sequence isAP EAMCET 2021 Medium