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

Statement-1: The middle term of (x+(1)/(x))^(2n) can exceed ((2n)^(n))/(n!) for some value of x. Statement-2: The coefficient of x^(n) in the expansion of (1-2x+3x^(2)-4x^(3)+ . . .)^(-n) is (1*3*5 . . .(2n-1))/(n!)*2^(n) . Statement-3: The coefficient of x^(5) in (1+2x+3x^(2)+ . . .)^(-3//2) is 2.1.

Statement-1: The middle term of (x+(1)/(x))^(2n) can exceed ((2n)^(n))/(n!) for some value of x. Statement-2: The coefficient of x^(n) in the expansion of (1-2x+3x^(2)-4x^(3)+ . . .)^(-n) is (1*3*5 . . .(2n-1))/(n!)*2^(n) . Statement-3: The coefficient of x^(5) in (1+2x+3x^(2)+ . . .)^(-3//2) is 2.1.

If x = 1/(2-sqrt3) then find the value of (x^(3) -2x^2 - 7x +4)

If |x| is so small that x^2 and higher powers of x many be neglected , prove that ((1 + 7x)^(2/3) . (1 - 4x)^(-2))/((4 + 7x)^(1/2)) = 1/2 (1 + 283/24 x)

If |x| is so small that x^2 and higher powers of x many be neglected , prove that ((1 + 7x)^(2/3) . (1 - 4x)^(-2))/((4 + 7x)^(1/2)) = 1/2 (1 + 283/24 x)

If sum_(r=1)^(n)r^(4)=F(x), then prove that the value of sum_(n=1)^(n)r(n-r)^(3) is (1)/(4)[n^(3)(n+1)^(2)-4F(x)]

If |x| is so small that x^2 and higher powers of x may be neglected show that ((4 - 7x )^(1//2))/((3 + 5x)^3) = 2/27 (1 - 47/8 x)

If |x| is so small that x^2 and higher powers of x may be neglected show that ((4 - 7x )^(1//2))/((3 + 5x)^3) = 2/27 (1 - 47/8 x)

If the number of solutions of sin^(-1)x+|x|=1cos^(-1)x+|x|=1,tan^(-1)x+|x|=1,cot^(-1)x+|x|=1,sec^(-1)x+|x|=1 and cos ec^(-1)are n_(1),n_(2),n_(3),n_(4),n_(5),n_(6) respectively,then then then then the value of n_(1)+n_(2)+n_(3)+n_(4)+n_(5)+n_(5) is