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 !)`

Prove that: s in x+s in3x++sin(2n-1)x=(sin^(2)nx)/(s in x) for all n in N.

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.

The coefficient of x^(n) in (1-(x)/(1!)+(x^(2))/(2!)-(x^(3))/(3!)+...+((-1)^(n)x^(n))/(n!))^(2) is equal to

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

Prove that the coefficient of x^(n) in the expansion of (1)/((1-x)(1-2x)(1-3x))is(1)/(2)(3^(n+2)-2^(n+3)+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)