is equal to:
Question:

$\lim _{x \rightarrow \infty}\left[\frac{1}{n}+\frac{n}{(n+1)^{2}}+\frac{n}{(n+2)^{2}}+\ldots \cdots+\frac{n}{(2 n-1)^{2}}\right]$ is equal to:

  1. (1) 1

  2. (2) $\frac{1}{3}$

  3. (3) $\frac{1}{2}$

  4. (4) $\frac{1}{4}$


Correct Option: , 3

Solution:

$\lim _{x \rightarrow \infty} \sum_{r=0}^{n-1} \frac{n}{(n+r)^{2}}=\operatorname{Lim}_{x \rightarrow \infty} \sum_{r=0}^{n-1} \frac{n^{2}}{n^{2}\left(1+\frac{r}{u}\right)^{2}}=\int_{0}^{1} \frac{d x}{(1+x)^{2}}$

$=-\left[\frac{1}{1+x}\right]_{0}^{1} \Rightarrow-\left[\frac{1}{2}-1\right]=\frac{1}{2}$

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