Question:
$\lim \left[\frac{1}{(\boldsymbol{w})}+\frac{\mathrm{n}}{(\mathrm{n}+1)^{2}}+\frac{\mathrm{n}}{(\mathrm{n}+2)^{2}}+\ldots \ldots+\frac{\mathrm{n}}{(2 \mathrm{n}-1)^{2}}\right]$
Correct Option: 1
Solution:
$\lim _{n \rightarrow \infty}\left[\frac{1}{n}+\frac{n}{(n+1)^{2}}+\frac{n}{(n+2)^{2}}+\ldots+\frac{n}{(2 n-1)^{2}}\right]$
$=\lim _{n \rightarrow \infty} \sum_{r=0}^{n-1} \frac{n}{(n+r)^{2}}=\lim _{n \rightarrow \infty} \sum_{r=0}^{n-1} \frac{n}{n^{2}+2 n r+r^{2}}$
$=\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{r=0}^{n-1} \frac{1}{(r / n)^{2}+2(r / n)+1}$
$=\int_{0}^{1} \frac{d x}{(x+1)^{2}}=\left[\frac{-1}{(x+1)}\right]_{0}^{1}=\frac{1}{2}$