By the definition of the definite integral, the value of _ n ( 1 n^2-1 + 1 n^2-2^2 + + 1 n^2-(n-1)^2 ) is equal to

Step-by-step solution

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