Prove that for every real number `[x+[x+1/n] +[x+2/n]+...+[x+(n-1)/n]=[nx]`

Prove that (1+x)^(n)>=(1+nx) for all natural number n,where x>-1

Prove that the term independent of x in the expansion of (x+(1)/(x))^(2n) is (1.3.5....(2n-1))/(n!)*2^(n)

Prove that the term independent of x in the expansin of (x+(1)/(x))^(2n) is (1.3.5(2n-1))/(n!)*2^(n)

If (lim_(x rarr0)({(a-n)nx-tan x}sin nx)/(x^(2))=0 where n is nonzero real number,the a is 0 (b) (n+1)/(n)( c) n(d)n+(1)/(n)

Let n be a natural and let 'a' be a real number. The number of zeros of x^(2n+1)-(2n+1)x+a=0 in the interval [-1,1] is -

Prove that nC_ (1) sin x * cos (n-1) x + nC_ (2) sin2x * cos (n-2) x + nC_ (3) sin3x * cos (n-3) x + ... + nC_ ( n) sin nx = 2 ^ (n-1) sin nn

Let n is a rational number and x is a real number such that |x|lt1, then (1+x)^(n)=1+nx+(n(n-1)x^(2))/(2!)+(n(n-1)(n-2))/(3!).x^(3)+ . . . This can be used to find the sm of different series. Q. The sum of the series 1+(1)/(3^(2))+(1*4)/(1*2)*(1)/(3^(4))+(1*4*7)/(1*2*3)*(1)/(3^(6))+ . . . . is

Let n is a rational number and x is a real number such that |x|lt1, then (1+x)^(n)=1+nx+(n(n-1)x^(2))/(2!)+(n(n-1)(n-2))/(3!).x^(3)+ . . . This can be used to find the sum of different series. Q. Sum of infinite series 1+(2)/(3)*(1)/(2)+(2)/(3)*(5)/(6)*(1)/(2^(2))+(2)/(3)*(5)/(6)*(8)/(9)*(1)/(2^(3))+ . . .oo is

If [x] denotes the greatest integer less than or equal to x and n in N , then f(X)= nx+n-[nx+n]+tan""(pix)/(2) , is