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$T h e v a l u e o f \lim_{n \to \infty} n [\frac{1}{(n + 1) (n + 2)} + \frac{1}{(n + 2) (n + 4)} + \dots + \frac{1}{6 n^{2}}]$

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Correct option is C

The given limit is L=limn→∞ ∑r=1n n⋅1(n+r)(n+2r) =limn→∞ 1n∑r=1n 11+rn1+2rn =∫01 dx(1+x)(1+2x) =∫01 −11+x+21+2xdx =[−log(1+x)+log(1+2x)]01 =[(−log2+log3)−(−log1+log1)] =log32

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The value of limn→∞ n1(n+1)(n+2)+1(n+2)(n+4)+⋯+16n2

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$T h e v a l u e o f \lim_{n \to \infty} n [\frac{1}{(n + 1) (n + 2)} + \frac{1}{(n + 2) (n + 4)} + \dots + \frac{1}{6 n^{2}}]$