The value of lim n → ∞ r + 2 r + . . . + n r n 2 , where r is non-zero real number and r denotes the greatest integer less than or equal to r , is equal to :

Step-by-step solution

We know that r ≤ r < r + 1 2 r ≤ 2 r < 2 r + 1 3 r ≤ 3 r < 3 r + 1 ⋮     ⋮       ⋮ n r ≤ n r < n r + 1 On adding all the above inequalities, we get r + 2 r + … + n r ≤ r + 2 r + . . . + n r < r + 2 r + . . . n r + n Using 1 + 2 + 3 + . . . + n = n n + 1 2 ,   n ∈ N , n n + 1 2 · r ≤ r + 2 r + . . . + n r < n n + 1 2 · r + n ⇒ n n + 1 2 · r n 2 ≤ r + 2 r + . . . + n r n 2