`sum_(n=0)^oo1/(n !)[sum_(k=0)^n(k+1)int_0^1 2^(-(k+1)x)dx]`

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

sum_(k =1)^(n) k(1 + 1/n)^(k -1) =

The value of int_(0)^(1)lim_(n rarr oo)sum_(k=0)^(n)(x^(k+2)2^(k))/(k!)dx is:

Suppose det [{:(sum_(k=0)^(n)k,,sum_(k=0)^(n).^nC_(k)k^2),(sum_(k=0)^(n).^nC_(k)k,,sum_(k=0)^(n).^nC_(k)3^(k)):}]=0 holds for some positive integer n. then sum_(k=0)^(n)(.^nC_(k))/(k+1) equals ............

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

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

The value of integral sum _(k=1)^(n) int _(0)^(1) f(k - 1+x) dx is