lim_(n rarr oo)(1)/(n^(2))sum_(k=0)^(n-1)[k int_(k)^(k+1)sqrt((x-k)(k+1-x))dx]

Evaluate: lim_ (n rarr oo) (1) / (n ^ (2)) sum_ (k = 0) ^ (n-1) [k int_ (k) ^ (k + 1) sqrt ((xk) (k + 1-x)) dx]

lim_(n rarr oo) 1/n^(3) sum_(k=1)^(n) k^(2)x =

Lt_(x to oo) (1)/(n^(3)) sum_(k=1)^(n)[k^(2)x]=

lim_ (n rarr oo) sum_ (k = 0) ^ (n) ((1) / (nC_ (k)))

lim_ (n rarr oo) n sum_ (k = 0) ^ (n-1) sum_ (k = 0) ^ (n-1) int _ ((k) / (n)) ^ ((k + 1) / ( n)) sqrt ((x- (k) / (n)) ((k + 1) / (n) -x)) dx is (pi) / (k) then k

Value of L = lim_ (n rarr oo) (1) / (n ^ (4)) [1sum_ (k = 1) ^ (n) k + 2sum_ (k = 1) ^ (n-1) k + 3sum_ ( k = 1) ^ (n-2) k + ...... + n.1] is

lim_ (n rarr oo) sum_ (k = 1) ^ (n) (k ^ (2)) / (2 ^ (k))

Find the value lim_(n rarr oo) sum_(k=2)^(n) cos^(-1) ((1 + sqrt((k -1) k(k + 1) (k + 2)))/(k(k + 1)))