Question

Prove that the coefficient of $x^r$ in the expansion of $(1-4x{)}^{-\frac{1}{2}}$ in $\frac{|\underset{–}{2r}|}{(|\underset{–}{r}|{)}^2}$

Answer 1

$(1+x{)}^n=1+nx+(n\left)\right(n-1)\frac{x^2}{2!}+......(n!)\frac{x^n}{n!}$

Where n is negative or fraction

given that

$(1-4x{)}^{\frac{-1}{2}}=1+(\frac{-1}{2})x+(\frac{-1}{2}\left)\right(\frac{-1}{2}-1)\frac{x^2}{2!}+......(\frac{-1}{2}\left)\right(\frac{-1}{2}-1)...(\frac{-1}{2}-r+1)\frac{x^r}{r!}$

coefficient of $x^r$ is

$\frac{(-1).(-3).(-5)....(1-2r)}{2^r.r!}$