`sum_(n=1)^oo1/(sqrt(n)+sqrt(n+1))`

Let S=sum_(n=1)^(9999)1/((sqrt(n)+sqrt(n+1)) (^4sqrt(n)+^4sqrt(n+1)) , then S equals ___________.

sum_(n=1)^oo sin^-1( (sqrt(n)-(sqrt(n-1)))/(sqrt((n)(n+1)) )= (A) pi/4 (B) pi/2 (C) - pi/3 (D) pi/3

sum_(n=1)^oo sin^-1( (sqrt(n)-(sqrt(n-1)))/(sqrt((n)(n+1)) )= (A) pi/4 (B) pi/2 (C) - pi/3 (D) pi/3

Suppose sum_(n=1)^(oo)(1)/((n+2)sqrt(n)+nsqrt(n+2))=(sqrt(b)+sqrt(c))/(sqrt(a)) where a,b,c in N and A = ((sqrt(a),b),(c,sqrt(a))) then (det(A))/(bc) is equal to ____

lim_(n rarr oo)(1)/(sqrt(n)sqrt(n+1))+(1)/(sqrt(n)sqrt(n+2))+......+(1)/(sqrt(n)sqrt(4n))

Let T_(n)=(1)/((sqrt(n)+sqrt(n+1))(4sqrt(n)+4sqrt(n+1))) and S_(n)=sum_(r=1)^(n)T_(r) then find S_(15) .

Let T_(n)=(1)/((sqrt(n)+sqrt(n+1))(4sqrt(n)+4sqrt(n+1))) and S_(n)=sum_(r=1)^(n)T_(r) then find S_(15) .

lim_(n->oo)sum_(r=1)^n(sqrt(n))/(sqrt(r)(3sqrt(r)+4sqrt(n))^2)

The value of sum_(k=2)^(oo){Lt_(n rarr oo)sum_(r=1)^(n)((sqrt(n))/(sqrt(r)(k sqrt(n)-sqrt(r))^(2))}]}

The value of ("lim")_(n rarr oo)sum_(r=1)^(4n)(sqrt(n))/(sqrt(r)(3sqrt(r)+sqrt(n))^2) is equal to