JEE Main2005MathematicsLimitsActual

_ n [ 1 n^2 ^2 1 n^2 + 2 n^2 ^2 4 n^2 + .+ 1 n^2 ^2 1 ] equals

Options

A1 2 1
B1 2 cosec 1
C1
D1 2 1

Correct answer

D. 1 2 1

Step-by-step solution

_ n [ 1 n^2 ^2 1 n^2 + 2 n^2 ^2 4 n^2 + 3 n^2 ^2 9 n^2 + .+ 1 n ^2 1 ] is equal to _ n r n^2 ^2 r^2 n^2 = _ n 1 n r n ^2 r^2 n^2 Given limit is equal to value of integral ₀^1 x ^2 x^2 d x or 1 2 ₀^1 2 x x^2 d x= 1 2 ₀^1 ^2 t d t [put .x^2=t ]= 1 2 ( t)₀^1= 1 2 1 .

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