JEE Main 2021 (Online) 25th February Evening ShiftQuestion 1
$\lim_{n\to\infty}\left[\frac{1}{n}+\frac{n}{(n+1)^{2}}+\frac{n}{(n+2)^{2}}+\ldots+\frac{n}{(2n-1)^{2}}\right]$ is equal to :
Limits & Continuity
Solution
Sum $=\frac1n+\sum_{k=1}^{n-1}\frac{n}{(n+k)^2}=\frac1n+\frac1n\sum_{k=1}^{n-1}\frac{1}{(1+k/n)^2}\to\int_0^1\frac{dx}{(1+x)^2}=1-\frac12=\frac12$. Option B.