If `f(x) = lim_(n->oo) sum_(r=0)^n (tan(x/2^(r+1)) + tan^3 (x/2^(r+1)))/(1- tan^2 (x/2^(r+1)))` then `lim_(x->0) f(x)/x` is

sum_(r=1)^(n)tan((x)/(2^(r)))sec((x)/(2^(r-1)));r,n in N

If f(x) = lim_(n->oo) tan^(-1) (4n^2(1-cos(x/n))) and g(x) = lim_(n->oo) n^2/2 ln cos(2x/n) then lim_(x->0) (e^(-2g(x)) -e^(f(x)))/(x^6) equals

Let f(x)=lim_(nrarroo) sum_(r=0)^(n-1)(x)/((rx+1){(r+1)x+1}) . Then

Let f (x) = lim_ (n rarr oo) sum_ (r = 0) ^ (n-1) (x) / ((rx + 1) {(r + 1) x + 1})

If f(x) = tan^(-1)((2^x)/(1 + 2^(2x + 1))) , then sum_(r = 0)^(9) f(r ) is

If f(x) = tan^(-1)((2^x)/(1 + 2^(2x + 1))) , then sum_(r = 0)^(9) f^(r ) is

lim_ (x rarr oo) sum_ (r = 0) ^ (n) ((1) / ((r + 2) n ^ (r)) =

value of lim_ (n rarr oo) sum_ (r = 1) ^ (n) tan ^ (- 1) ((1) / (2r ^ (2))) is