If _ n [ (1+ 1 n^2 ) (1+ 2^2 n^2 ) (1+ n^2 n^2 ) ]^ 1 / n =k then k=

Step-by-step solution

Given, aligned & k= _ n [ (1+ 1 n^2 ) (1+ 2^2 n^2 ) (1+ n^2 n^2 ) ]^ 1 / n & k= _ n [ (1+ 1 n^2 ) (1+ 2^2 n^2 ) (1+ n^2 n^2 ) ]^ 1 / n = & _ n 1 n _ r=1 ^n (1+ r^2 n^2 )= ₀^1 (1+x^2 ) d x = & [ (1+x^2 ) x ]₀^1- ₀^1 2 x 1+x^2 x d x = & 2-2 ₀^1 x^2 1+x^2 d x= 2-2 ₀^1 (1- 1 1+x^2 ) d x = & 2-2 [x- ⁻¹ x ]₀^1 = & 2-2 [1- 4 ] = & 2-2+ 2 = & 2+ 2 -2 aligned