$(20$ points$)$ Write $f(x)=\frac{1}{1+2x}$ as a Maclaurin series using two distinct methods. Choose from using $1)$ the Maclaurin series formula, $2)$ a known series, or $3)$ the binomial series. Be sure to find the radius of convergence.

$($a$)$ Method $1$

$($b$)$ Method $2$