If a, b, c R⁺ then Lim _ n _ k=1 ^n n (k+a n)(k+b n) is equal to

Step-by-step solution

We have Lim _ n _ k=1 ^n n n^2 (a+ k n ) (b+ k n ) = Lim _ n 1 n _ k=1 ^n 1 (a+ k n ) (b+ k n ) = ₀^1 1 (a+x)(b+x) d x= 1 a-b ₀^1 (a+x)-(b+x) (a+x)(b+x) d x= 1 a-b ₀^1 ( 1 b+x - 1 a+x ) d x= 1 a - b [ ( b + x )- ( a + x )]₀^1= 1 a - b [ ( b + x ) ( a + x ) ]₀^1= 1 a - b ( b +1) a ( a +1) b (B) Now if a = b then .I= Lim _ n _ k =1 ^ n 1 n 1 ( a + k n )^2 = ₀^1 d x ( a + x )^2 = 1 a + x ]₁^0= 1 a - 1 a +1 = 1 a ( a +1) D