Prove that the coefficient of `x^r` in the expansion of `(1-2x)^(-1/2)` is `(2r!)/[(2^r)(r!)^2]`

The coefficient of x^(r) in the expansion of (1-x)^(-2) is

Prove that the coefficient of x^r in the expansion of (1-2x)^(1//2)i s(2r)!//[2^r(r !)^2]dot

the coefficient of x^(r) in the expansion of (1 - 4x )^(-1//2) , is

the coefficient of x^(r) in the expansion of (1 - 4x )^(-1//2) , is

The coefficient of x^(r) in the expansion of (1-ax-x^(2))/(e^(x)) is

If P_(r) is the coefficient of x^® in the expansion of (1 + x)^(2) (1 + (x)/(2))^(2) (1 + (x)/(2^(2)))^(2) (1 + (x)/(2^(3)))^(2)... prove that P_(r) = (2^(2))/((2^(r) -1))(P_(r-1) + P_(r-2) )and P_(4) = (1072)/(315) .

If |x|<(1)/(2), then the coefficient of x^(r) in the expansion of (1+2x)/((1-2x)^(2)) is

If a_r is the coefficient of x^r in the expansion of (1-2x+3x^2)^n then sum_(r=0)^(2n) r a_r =