Let S=`sum_(n=1)^(9999)1/((sqrt(n)+sqrt(n+1))(root4(n)+root4(n+1)))` , then S equals ___________.

If the value of lim_(n rarr oo)(n^(-3/2)) sum_(j=1)^(6n) sqrt(j) is equal to sqrt(N) , then the value of N/(12) is __________

If sum_(r=1)^(r=n)(r^(4)+r^(2)+1)/(r^(4)+r)=(675)/(26) , then n equal to

The value of ("lim")_(nvecoo)sum_(r=1)^(4n)(sqrt(n))/(sqrt(r)(3sqrt(r)+sqrt(n))^2) is equal to 1/(35) (b) 1/4 (c) 1/(10) (d) 1/5

lim_(nrarroo) (1-x+x.root n e)^(n) is equal to

If sum_(r=0)^(n){("^(n)C_(r-1))/('^(n)C_(r )+^(n)C_(r-1))}^(3)=(25)/(24) , then n is equal to

The value of sum_(i=1)^(13) (n^(n) + i^(n-1)) is

sum_(r=1)^nsin^(-1)((sqrt(r)-sqrt(r-1))/(sqrt(r(r+1)))) is equal to (a) tan^(-1)(sqrt(n))-pi/4 (b) tan^(-1)(sqrt(n+1))-pi/4 (c) tan^(-1)(sqrt(n)) (d) tan^(-1)(sqrt(n)+1)

sum_(n=1)^ootan(theta/(2^n))/(2^(n-1)cos(theta/(2^(n-1))))

If S_n=[1/(1+sqrt(n))+1/(2+sqrt(2n))+....+1/(n+sqrt(n^2))] then (lim)_(n ->oo)S_n is equal to (A) log 2 (B) log4 (C) log8 (D) none of these