JEE Main 2024 (Online) 30th January Morning ShiftQuestion 1
The value of $\lim_{n\to\infty}\sum_{k=1}^{n}\frac{n^{3}}{\left(n^{2}+k^{2}\right)\left(n^{2}+3k^{2}\right)}$ is :
Limits & Continuity
Solution
Concept. Write the sum as a Riemann sum, then use partial fractions. Formulas used. (1) $\lim\frac1n\sum_{k=1}^n\phi\big(\frac kn\big)=\int_0^1\phi(x)\,dx$(2) $\int_0^1\frac{dx}{1+c^2x^2}=\frac1c\tan^{-1}c$Step 1 - Riemann form. Divide top and bottom by $n^4$: $$\frac{n^3}{(n^2+k^2)(n^2+3k^2)}=\frac1n\cdot\frac1{(1+t^2)(1+3t^2)},\ t=\tfrac kn$$ By (1), $L=\int_0^1\frac{dx}{(1+x^2)(1+3x^2)}$. Step 2 - Partial fractions. Put $y=x^2$: $1=A(1+3y)+B(1+y)$. $y=-1$: $A=-\frac12$; $y=-\frac13$: $B=\frac32$. Step 3 - Integrate. By (2): $$L=-\frac12\cdot\frac\pi4+\frac32\cdot\frac1{\sqrt3}\cdot\frac\pi3=\frac{(4\sqrt3-3)\pi}{24}$$ Step 4 - Match option. $\frac{13\pi}{8(4\sqrt3+3)}=\frac{13\pi(4\sqrt3-3)}{8\cdot39}=\frac{(4\sqrt3-3)\pi}{24}$. Answer: D