_ n 1 n [1 + 1 2 + 1 3 + + 1 n ]

Step-by-step solution

The given limit is _ n 1 n [ 1 + 1 2 + 1 3 + + 1 n ] . This can be written in summation notation as _ n 1 n _ r=1 ^ n 1 r . To convert this into a Riemann sum, multiplying and dividing the expression by n gives _ n 1 n _ r=1 ^ n n r . This simplifies to _ n 1 n _ r=1 ^ n 1 r n . Converting the summation into a definite integral using the limit of a sum _ n 1 n _ r=1 ^ n f ( r n ) = ₀¹ f(x) dx , we obtain ₀¹ 1 x dx . Evaluating the definite integral gives ₀¹ x^ -1/2 dx = [ 2x^ 1/2 ]₀¹ = 2(1) - 2(0) = 2 . Therefore,