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The value of
$lim_{n \to \infty} (\frac{1}{n} + \frac{n}{(n + 1)^{2}} + \frac{n}{(n + 2)^{2}} + \dots + \frac{n}{(2 n - 1)^{2}})$ is

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Correct option is C

Required limit −limn→∞ ∑r=0n−1 n(n+r)2=limn→∞ 1n∑r=0n−1 11+rn2=∫01 dx(1+x)2=−11+x01=−12+1=12

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The value oflimn→∞ 1n+n(n+1)2+n(n+2)2+⋯+n(2n−1)2 is

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The value of
$lim_{n \to \infty} (\frac{1}{n} + \frac{n}{(n + 1)^{2}} + \frac{n}{(n + 2)^{2}} + \dots + \frac{n}{(2 n - 1)^{2}})$ is