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JEE Previous Year Question

$lim x \to \infty (\frac{1}{n} + \frac{n}{( n + 1 ) ^{2}} + \frac{n}{( n + 2 ) ^{2}} + \dots \dots + \frac{n}{( 2 n - 1 ) ^{2}}) is equal to$

Topic: JEE PYQ

Answer: C. Full step-by-step solution below.

A $1$
B $\frac{1}{3}$
C $\frac{1}{2}$
D $\frac{1}{4}$

Correct answer: C

Solution

c JEE Main 2021 lim x \to \infty \sum \frac{n}{( n + r ) ^{2}} = Lim_{x \to \infty} \sum \frac{n ^{2}}{n ^{2} ( 1 + \frac{r}{n} ) ^{2}} = \sum \frac{dx}{( 1 + x ) ^{2}} = - (\frac{1}{1 + x})_{0}^{1} \Rightarrow - (\frac{1}{2} - 1) = \frac{1}{2}

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