Let `x_1, x_2, ,x_n` be positive real numbers and we define `S=x_1+x_2++x_ndot` Prove that `(1+x_1)(1+x_2)(1+x_n)lt=1+S+(S^2)/(2!)+(S^3)/(3!)++(S^n)/(n !)`

If x,y are positive real numbers and m, n are positive integers, then prove that (x^(n) y^(m))/((1 + x^(2n))(1 + y^(2m))) le (1)/(4)

If x_1,x_2,x_3,... are positive numbers in G.P then logx_n, logx_(n+1), logx_(n+2) are in

If x and y are positive real numbers and m, n are any positive integers, then prove that (x^n y^m)/((1+x^(2n))(1+y^(2m))) lt 1/4

Prove that cos^(-1) ((1 - x^(2n))/(1 + x^(2n))) = 2 tan^(-1) x^(n), 0 lt x lt oo

Statement-1: If x, y are positive real number satisfying x+y=1 , then x^(1//3)+y^(1//3)gt2^(2//3) Statement-2: (x^(n)+y^(n))/(2)lt((x+y)/(2))^(n) , if 0ltnlt1 and x,ygt0.

If x_n =cos pi/(2^n) +isin pi/(2^n) , prove that x_1x_2x_3 x_oo=-1.

Prove that 1-^n C_1(1+x)/(1+n x)+^n C_2(1+2x)/((1+n x)^2)-^n C_3(1+3x)/((1+n x)^3)+....(n+1) terms =0

If x_1, x_2, x_3,...... x _(2n) are in A.P , then sum _(r=1)^(2n) (-1)^(r+1) x_r^2 is equal to (a) (n)/((2n-1))(x _(1)^(2) -x _(2n) ^(2)) (b) (2n)/((2n-1))(x _(1)^(2) -x _(2n) ^(2)) (c) (n)/((n-1))(x _(1)^(2) -x _(2n) ^(2)) (d) (n)/((2n+1))(x _(1)^(2) -x _(2n) ^(2))

If S.D of x_(1), x_(2), x_(3),….x_(n),… is sigma , then find S.D of -x_(1), -x_(2), -x_(3),….-x_(n) ?

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