" 4.The value of "Lt(x+x^(2)+x^(3)+....+x^(n)-n)/(x-1)" is "

underset(x to 1)"Lt" (x+x^(2)+...+x^(n)-n)/(x-1)=

If x^(2)-x+1=0 then the value of sum_(n=1)^(5)[x^(n)+(1)/(x^(n))]^(2) is:

If lim _(xto1)(x^(1)+x^(2)+x^(3)+"..."+x^(n)-n)/(x-1)=820,(nin N) then the value of n is equal to

The value of lim_(x rarr1)(x^(n)+x^(n-1)+x^(n-2)+...x^(2)+x-n)/(x-1)

The value of ""(n)C_(1). X(1 - x )^(n-1) + 2 . ""^(n)C_(2) x^(2) (1 - x)^(n-2) + 3. ""^(n)C_(3) x^(3) (1 - x)^(n-3) + ….+ n ""^(n)C_(n) x^(n) , n in N is

The value of ""(n)C_(1). X(1 - x )^(n-1) + 2 . ""^(n)C_(2) x^(2) (1 - x)^(n-2) + 3. ""^(n)C_(3) x^(3) (1 - x)^(n-3) + ….+ n ""^(n)C_(n) x^(n) , n in N is

If | x| lt 1, then the coefficient of x ^(n) in (1 + 2x + 3x ^(2)+ 4x ^(3) +...) ^(1//2), is :

Neglecting x^(n) for n>=2, value of (1-7x)^((1)/(3))(1+2x)^(-(3)/(4)) is

For each ositive integer n, define a function f _(n) on [0,1] as follows: f _(n((x)={{:(0, if , x =0),(sin ""(pi)/(2n), if , 0 lt x le 1/n),( sin ""(2pi)/(2n) , if , 1/n lt x le 2/n), (sin ""(3pi)/(2pi), if, 2/n lt x le 3/n), (sin "'(npi)/(2pi) , if, (n-1)/(n) lt x le 1):} Then the value of lim _(x to oo)int _(0)^(1) f_(n) (x) dx is:

For each ositive integer n, define a function f _(n) on [0,1] as follows: f _(n((x)={{:(0, if , x =0),(sin ""(pi)/(2n), if , 0 lt x le 1/n),( sin ""(2pi)/(2n) , if , 1/n lt x le 2/n), (sin ""(3pi)/(2pi), if, 2/n lt x le 3/n), (sin "'(npi)/(2pi) , if, (n-1)/(n) lt x le 1):} Then the value of lim _(x to oo)int _(0)^(1) f_(n) (x) dx is: