Prove the following

Question:

$\lim \left[\frac{1}{(\boldsymbol{w})}+\frac{\mathrm{n}}{(\mathrm{n}+1)^{2}}+\frac{\mathrm{n}}{(\mathrm{n}+2)^{2}}+\ldots \ldots+\frac{\mathrm{n}}{(2 \mathrm{n}-1)^{2}}\right]$

  1. $\frac{1}{2}$

  2. 1

  3. $\frac{1}{3}$

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


Correct Option: 1

Solution:

$\lim _{n \rightarrow \infty}\left[\frac{1}{n}+\frac{n}{(n+1)^{2}}+\frac{n}{(n+2)^{2}}+\ldots+\frac{n}{(2 n-1)^{2}}\right]$

$=\lim _{n \rightarrow \infty} \sum_{r=0}^{n-1} \frac{n}{(n+r)^{2}}=\lim _{n \rightarrow \infty} \sum_{r=0}^{n-1} \frac{n}{n^{2}+2 n r+r^{2}}$

$=\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{r=0}^{n-1} \frac{1}{(r / n)^{2}+2(r / n)+1}$

$=\int_{0}^{1} \frac{d x}{(x+1)^{2}}=\left[\frac{-1}{(x+1)}\right]_{0}^{1}=\frac{1}{2}$

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